Go
HERE for campground reviews/recommendations from many fellow RVers. There are four pages of RV parks and campgrounds listed on this Website just for the West Yellowstone area, far too many to review them on this thread.
I have worked in Yellowstone, staying in West Yellowstone and commuting to the Norris Geyser Basin. As others have said, even that relatively short distance can be very long during the early morning "rush hour" of tourists day-tripping from the west side of the park.
If I were doing your trip, I would try to get a reservation at relatively centrally located Canyon Village or a fc-fs site at Norris. Then you could visit Yellowstone Falls and Hayden Valley one day and the Upper and Lower Geyser Basins the following day. And, if you came north from Grand Teton NP, you would pass Yellowstone Lake on your way to Canyon. So, you would have seen most of the major attractions of YNP and the spectacular Tetons as well in only three looong days of driving.