My initial thought was something regarding the High Sparrow's identity that they could use against him… not necessarily HR=HS, but obviously that would be in play. I like the wildfire idea, too.

That's because Stannis took ships, and docked near the Wall. I guess the armies of the Vale could sail - I have no idea what their fleet situation would look like, or where they could dock safely and without drawing attention.

Actually, they don't need to go through the Twins, but definitely Moat Cailin. I would guess the Boltons are still treating him as an ally, as was stated above