a) The Starks are known for their honour, I doubt any of them would have even considered this an option.

b) There are 2 people that Robin Arryn (Current Lord of the Vale) trusts left on Westeros.  Yohn Royce, and Sansa Stark.  And as everyone has shown, Robin is very easy to manipulate, so in essence, they already control the Vale. (Yohn was present at the execution of Littlefinger.)

c) By declaring for House Stark during the Battle of the Bastards, Littlefinger burned many of his bridges and revealed himself to the world as a deceitful schemer.  His name wouldn't carry the same weight after that.