There's two KOA campgrounds near West Yellowstone. I stayed at the one closer to the town of West Yellowstone, which is a much nicer of the two.
http://yellowstonekoa.com/Warning though, the KOA isn't cheap during prime tourist season.
Reservations are surely required there, or any of the in-park campgrounds that offer reservations. I've heard that the non reservation campgrounds in the park fill up by 9am.
There's a few other campgrounds in the town of West Yellowstone, but I didn't investigate them when I was there in September.
West Yellowstone makes a nice base camp because when it comes time to restock on groceries, there's a few stores in town.