Even though the Avengers in the Cinematic Universe have had sparse romantic relationships, their same counterparts in the comics have been around for decades and have been fairly promiscuous! Here's a list of the most notable Avengers romances.
Peggy Carter - Obviously Captain America's most popular love interest is his 1940s love. We know her well as she's become a staple in the MCU, and even though these two had a brief and casual romance, Peggy was very influenced by Steve's character and their brief time together.
Scarlet Witch - Although the two's romance was brief, it's still considered a very odd pairing: someone who is totally straight laced with a former villain and constantly unbalanced psyche. The two must have bee interesting to write because of their differences, but in my opinion they had no chemistry or anything in common to keep them together.
Sharon Carter - After Cap woke up from his deep freeze, he and Sharon crossed paths several times over their hero-ing, he with the Avengers and she with S.H.I.E.L.D. After the superhero Civil War Cap was facing charges of treason and Sharon, though a victim of mind control, is the sniper who "kills" Cap on his way to trial. Eventually it's shown that his death was faked by Red Skull (who was also the one behind controlling Sharon), and when he comes back he and Sharon have reconciled again. Sharon is considered by fans to be Cap's greatest love, and, as we will be seeing more of her in Cap 3 next year, it's likely we'll see this play out on screen.
Wasp - Usually in the comics Janet is romantically involved with her off/on again husband Hank Pym, but during one of their separations (after he beat her up- yeah Marvel had a hero beat his wife...) she and Stark had a short lived romance.
Pepper Potts - We know that Tony Stark is supposed to end up with his beautiful assistant, and later CEO, of Stark Industries in the MCU, but did you know that originally in the comics she was the middle of a love triangle between Stark and his bodyguard/chauffer/assistant Happy Hogan? Hogan appeared in all three Iron Man movies and you would for sure never guess from just the films that this was their history! Potts even picked Hogan over Stark, but ended up leaving him several years later. In the comics, Stark and she reconcile and end up in an off/on again relationship. He even gives her some armor and she takes on the codename Rescue and joins the Avengers.
Betty Ross/Red She-Hulk - Ross and Banner were romantically connected before he ever became the Hulk. Eventually he went on the run and the two did not stay in contact. In the meantime Ross' father, "Thunderbolt Ross," turned himself into the Red Hulk, and eventually Betty was turned into the Red She-Hulk, and even though Hulk and Banner have both had their own romances Betty has been the one constant in all variations and continuities of Hulk's story and she's usually the only person who can bring Banner back from the Hulk.
Caiera - There was a time in the comics where Hulk was so powerful and uncontrollable that several of our heroes got together and decided he needed to be banished before he did something completely terrible, so they launched him into space thinking he'd be secluded and couldn't hurt anyone. Instead he landed on a planet where he quickly became a slave and sold into gladiator contests. Eventually Hulk rose up and led a rebellion against the monarchy, and Caiera turns on her King, seeing how awful he truly is. The couple have two children: Hiro-Kala and Skaar before she is killed and Hulk leaves back for Earth, believing their children also dead.
Iron Man - In the Ultimate Marvel comic continuity the Ultimates (their form of the Avengers) Widow and Stark had a long romance, and were even engaged. Instead of an engagement ring, Natasha received a suit of black armor. She ended up betraying Stark and all the Ultimates, and killed Edwin Jarvis and Hawkeye's wife and children. (Yes, the same family we saw in Avengers: Age of Ultron.) In the end end, Hawkeye snuck into her prison cell and assassinated her in revenge for his family.
Daredevil - After leaving both S.H.I.E.L.D. and the Avengers for some independent superhero work, she teams up with Matt Murdock; but ultimately the two break up because the relationship is going nowhere and Natasha again feels like she is playing the sidekick.
Winter Soldier - In the mainstream comics, these two are considered each other's most prominent relationship, and I would love to see this translate back to the big screen now with Hulk out of the picture. These two could have so much in common with their shared brainwashing and attempts at being heroes after being villains, and we could see some really cool things happen if Bucky became the next Captain America.
Enchantress - Even though she's one of Thor's most prominent villains, they have been romantically linked a time or two--but usually not by Thor's own choice. In the comics and the cartoon Earth's Mightiest Heroes the Enchantress is enamored with Thor and attempts to use her magics to seduce him several times.
Lady Sif - It was hinted at in the Thor films that Thor's father Odin was hoping he'd hit it off with Lady Sif in Asgard, even though this didn't play out in the movies, their comic counterparts have had a romantic relationship that was complicated with all of his traveling and superhero "work" on Earth, and it eventually took a toll on their relationship.
Jane Foster - Thor's most prominent love interest shouldn't come as a surprise to anyone, and we've seen her in two Thor films to date. In the comics she's a nurse that Thor meets as his alter ego Dr. Donald Blake, Thor's secret identity. Eventually he shares that he's the God of Thunder with her and the two have yet another cliche comic book on and off again romance. Eventually when they're off Jane is diagnosed with cancer and living with her new boyfriend when Thor drops in. He insists on finding a way to use magic to cure her, but she refuses and battling cancer Jane becomes the new Thor and wielder of his hammer during a time where Thor has been unworthy to hold it.
Black Widow - In the mainstream comic continuity, Widow fell in love with Hawkeye and that was her base for leaving the KGB behind and going straight with the Avengers. Her former employers tried to have her killed and Hawkeye saved her life, even though the relationship never went anywhere.
Mockingbird - Bobbi has been a long on and off again relationship for Barton, and is considered his strongest romance in the main Marvel continuity. When it began the two simply fell in love through a team up and were married shortly afterward. The marriage didn't last long as Bobbi was brainwashed and mind controlled into sex with a villain, whom she killed when she came to her senses. Hawkeye was not only hurt by her betrayal, but angry she let the man die before facing trial. In the end, the pair reconcile, but she is soon killed in a battle and dies in his arms.
However it turns out that this was a Skrull sleeper agent who had been posing as Mockingbird and during the Skrull Invasion of Earth it was revealed that she was alive and well, but had been held captive by the aliens for years and she and Clint rebooted their romance.
Fans were hoping that we'd see this romance come to life in the MCU, and rumors swirled after Bobbi joined the cast of Agents of S.H.I.E.L.D., but in the newest Avengers film we saw Hawkeye happy with his wife and three children. It's unlikely they'll take the dark route in killing them like they did in the Ultimate comics so unfortunately it's unlikely we'll see Mockingbird and Hawkeye together.
Wonderman - When Hank Pym created Ultron, and Ultron turned around and created Vision he used Wonderman's basic brain patterns as a template for the android. So it was only natural when Vision was destroyed that Wonderman expressed his own feelings for Wanda, which they deduced was where Vision's original romantic feelings stemmed from. In the end, Wonderman lent his brain patterns to the Vision a second time as he was rebuilt and their short term relationship ended.
Nightcrawler - In the comics, Wanda started out as a villain under the leadership of mutant terrorist Magneto. In the cartoon Wolverine and the X-Men, both Nightcrawler and Wanda have recurring roles, and they have an awkward long distance my-father-hates-you-and-the-X-Men relationship. Eventually Wanda sides with the X-Men.
Vision - Originally the two were members of the West Coast Avengers team where they fell in love without the blessing of her brother Quicksilver, and Hawkeye who was also in love with Wanda.
Eventually with Wanda's reality warping abilities that alter reality as she wishes it, she creates two children for her and Vision because they couldn't have them naturally. When Vision is dismantled, Wanda temporarily battles her own demons, so with the help of Dr. Strange he magically erases Wanda's children and her memories of them from existence since they weren't "real" anyway. The Vision is rebuilt, but the new version has no emotion and does not love his wife anymore. Wanda's distress continues and she turns to Doctor Doom for help this time, which makes her magic even more unstable. Wanda unknowingly attacks the Avengers at their mansion with villains that she's created from her magic. At the end of the attack, the mansion is destroyed, Hawkeye and Scott Lang are killed, Vision seems to betray the team and is killed by teammate She-Hulk. The attack only ends because Dr. Strange intervenes to stop Wanda. Magneto, her father, shows up and takes her to Xavier in hopes that he can help clear her mind.
After the Vision was rebuilt again and the Scarlet Witch came out of hiding to rejoin the Avengers to fight against the Phoenix cosmic force the two share a reunion that is less than positive, Vision snapped at Wanda, blaming her for his death and refused to be a part of a team that she was on.
Scarlet Witch - See Scarlet Witch
Stature - Vision was "reborn" because the Young Avengers put him back together after Wanda lost control of her powers when he was destroyed. Vision stuck around with this young team as a mentor of sorts, and eventually left with Stature (Scott Lang's daughter) to join an alternate team of Avengers because he was in love with her. The two had a short romance until she was killed and Vision was destroyed. The Young Avengers considered putting him back together, but they lacked the tech to do so, or a current backup of his memories and were fearful of filling him in on Stature's death. Instead they like to believe that the two are somewhere together.
Captain Marvel - Carol received her powers partly due to her encounter with alien Captain Marvel from the Kree when she was a Captain in the air force. Several different iterations of their encounter have been written and re-written for comics and cartoons, but recently in the cartoon Earth's Mightiest Heroes, Captain Marvel was researching humans in disguise and the encounter between him, his enemy, and Carol gave her her abilities. Marvel was imprisoned temporarily, but liberated to help the Avengers fight a larger foe, and the two have had a romantic relationship in several alternate realities and continuities.
Wonderman - The two have an on and off again relationship in the comics, and right before the Superhero Civil War the two were on again when they both sided with Tony Stark on the Registration's side. Ms Marvel became the leader of their splinter Avengers team after the Civil War had ended, but it hasn't been stated if the two are still together.
While I think either of Carol's loves would be a great addition to the upcoming Captain Marvel movie, Marvel has already announced I think "Why choose one? Bring in both characters and a love triangle!" Both Captain Marvel and Wonderman are such rich characters with so much history it would be really awesome to see both come to the big screen!
CAPTAIN AMERICA AND...IRON WOMAN?!?!??!?!
In an alternate universe somewhere labelled "Earth-3490" Tony Stark was born a woman instead of a man and named Natasha Stark. In their universe, the Superhero Civil War was avoided completely because Cap and Iron Woman were married and talked through their personal beliefs to come to create the 50 State Initiative. Instead of constantly bickering about their differences and viewpoints, the couple was able to do great things to not only eradicate the super villains in their world but to do much in the field of science.
There you have it. The sexual history of the Avengers! How many do you think we'll see come to play in the MCU?