As the current season of Game Of Thrones races towards a dramatic conclusion, Littlefinger, a.k.a. Lord Petyr Baelish, looks to be back to his manipulative best. This is never a good thing for those around him, particularly when those around him include our beloved Stark family. However, at least one Stark has the ability to bring Littlefinger down for good.
A Tenuous Grip On The Vale
At present, the main reason the Starks need to keep Littlefinger around is his control over the allegiance of the Knights of The Vale, apparently acting as Regent for the underage Lord of The Vale, Robin Arryn. His right to assume this regency relies on his short marriage to Robin's widowed mother — and sister to Catelyn Stark — Lysa Arryn. The marriage was short because Littlefinger murdered Lysa by pushing her through the Moon Door at the Eyrie, later claiming that she had jumped. While the Vale's lesser Lords doubted Littlefinger's version of events, they had little choice but to believe him after listening to the testimony of the only other witness to Lady Lysa's death.
Sansa is the only witness to the crime. While she lied in order to protect Littlefinger (who seemed to be her protector at that point), she could reveal the truth at any time. The Lords already distrust Littlefinger, so they could easily believe he had frightened or manipulated her into lying. The reveal of Baelish's role in Lysa's death would shatter his claim to regency over Robin, and with it any control he holds over the Knights of the Vale. The only thing currently stopping Sansa from doing so is the possibility of losing the allegiance of The Vale, who proved a decisive factor in Jon Snow's victory in the Battle of The Bastards. However, that may not be an issue.
Lady Regent Of The Vale
Robin Arryn is still currently underage, and will be for a few more years. So, even if Littlefinger were to lose his position as Robin's regent, another would need to be appointed. Usually, the position of Regent would fall to the underage Lord's closest living of-age relative. This, would be Robin's eldest surviving cousin: Sansa Stark.
Sansa, in her current role as acting Queen In The North, has already been gaining the respect of one of the Vale's major Lords, Yohn Royce, which would aid and ease any transition into her taking temporary leadership over The Vale. If this does come to pass, Sansa would have authority to keep the Knights of The Vale in the North for as long as they are needed.