The farthest-south CG in YNP that can accommodate RVs is Grant Village (no hookups); otherwise for the "south end" you would need to camp outside the park at Flagg Ranch (since renamed - not sure of the name of it now) or (better, IMO) Colter Bay RV Park in Grand Teton National Park. That makes for a long drive each day into the park. East entrance has a similar problem - a long drive into the heart of YNP from anywhere you can stay outside of it; the farthest-east CG inside YNP is Fishing Bridge (not "flying bridge"), which is an RV park, FHUs but very tight. Great for convenience but not for ambience, IMO.
The park is enormous - you'll spend an entire day just hitting the briefest of highlights for each of the two parts of the Grand Loop road in the park. Plan to spend several days in the park and base-camp in one or two spots. It will take you several hours to get between CGs because of traffic and wildlife (i.e. buffalo in the road, etc.), so plan accordingly. You could consider a base camp on the west or south sides (Colter Bay, Grant Village or Grizzly are good choices) for a few days and then on the north side (Mammoth - no hookups - or a private CG such as Rocky Mt RV Park in Gardiner, just outside the north entrance) for a few days, spending 1-3 days exploring each part of the Grand Loop. Another possibility is Fishing Bridge for your entire stay, since it's pretty centrally located - but again, no ambience, very tight sites. And should a car rental be needed, there are no rentals within the park - just in West Yellowstone, Gardiner, Jackson or Cody.
All entrances into the park except the northeast are RV-friendly, but some are more challenging than others. West entrance is by far the least mountainous driving, but all (N, S, E & W) are do-able. Just take it easy, and the driver may need to forego sightseeing in order to improve concentration on the road. :) People do drive through the park in big rigs with toads or trailers, but I don't recommend that except as a way to get to whatever CG you're planning to stay in. Otherwise, use your toad, tow vehicle or a rental car for the sightseeing.