If I remember correctly, the Vale forces wanted nothing to do with the conflict in Winterfell. But, because Robin Arryn was too immature to take any interest, he delegated that decision to Littlefinger.
For House Arryn, having Winterfell run by House Bolton or Stark loyalists didn't make enough of difference to go to battle?
It was Littlefinger's decision alone to do it (with Sansa encouragement)?
As of now, it appears Littlefinger has mostly unchecked control of the Vale army?