My chickens are in a slab-floored building, but about half the pens are up on OSB or plywood with vinyl on it (not done by me); others are on OSB on concrete, or just on concrete with lotsa bedding. All three methods work about equally well IMHO, I really do not subscribe to the "linoleum cult" around here LOL not that there is anything too much wrong with it (except for the potential for trapping dampness behind it, and the expense/labor) but to me just a good paint job on yer plywood is just as good. Mind, this is with a considerable amount of bedding on it. If I were going to try to skinch on bedding and knew there would often be bare spots on the floor, vinyl *might possibly* have an advantage.
Drafts don't go thru plywood, though ; if you have gaps at seams, caulk and cover them.
Good luck, have fun,
Pat