There is a nice campground at the north entrance to Yellowstone, Mammoth Hot Springs
which has water and restrooms - no hook-ups. It is first come, first serve. It is the only campground open year round, and was pretty empty when I was there over Christmas a couple of years ago.
If you want hook-ups and reservations, there is a nice place right outside the park in Gardiner
Rocky Mountain Campground
It was clean, and the people running it were friendly. They also have three little cabins on the propery, inexpensive, which a bunch of us rented one August, as my son got married at Mammoth Hot springs when we were there. Camping without the tent or the trailer..but a nice place to get ready for a wedding. Beautiful view above the river.