V isn't comfortable with the horses. Granted he was by nature rather scrawny and delicate compared to the other Sparda but there was something off about them that has him shying away from actually tending one. So for the most part he keeps to the wagons, wrapped in a warm blanket while his Shadow uses this chance to go running around in the snow and harassing other people and animals.

Which means that Shadow would definitely shadow a rider and horse, making little encouraging chirps or playfully pawing at the ground next to them. When camp is made, Shadow definitely has swiped at some food or if it's too well-guarded, edge into view and plaintively try the best woeful look that a big cat can muster while watching the food. There might even be slight little chirping sounds.

But since Shadow is not suppose to be wandering around bothering people, V will finally show up to corner the big cat and apologize. Otherwise, he is going to stay almost by himself, reading his book and acting like he isn't as nervous as everyone else. By the second day, he's taken to reading out loud bits of poetry though he may be talked into explaining the kind of demons they can find once they reach the Qliphoth.