The Plateau is high in elevation. The snow can get serious after Labor Day. October is late for Yellowstone. There should be some spaces available but you have to think of it as a winter type trip.
When National Parks get quiet in the late fall, they stop taking reservations for most campsites that are still open because the demand falls off rapidly. I like the off-season best. Off season in Yellowstone can have a lot of snow by October. It can be really cold by then, it is the luck of the draw. Cold as in zero of even lower.