You have several nice CG's to stay at. On your side, but close is Rex Hale CG - here's a link from some full-timer's currently staying there -
click here.Then as mentioned, and one of our favorites is West Yellowstone - you'll have to go in the East Entrance then back out the West side. If you like the resort style - stay at Grizzly RV in town. It's very nice. They have laundry and free coffee every morning in their lounge. Nice areas to walk your dogs too if you have any. Just 2 miles north of town is Baker's Hole - a FS CG with electric hookups at most sites and water spigots throughout to fill your tank. A threaded one at the entrance, the rest you'll need a Water Thief. This is a big rig friendly FS CG and one that we've stayed at several years in a row and quite a few sites are satellite friendly if you have one of those too. If you stay here and want to see The Grand Tetons but get closer, Colter Bay CG off of Jackson Lake has both a full hookups side and a dry camping side that fit rigs of any size.