Here are some photos to entice you -
Buffalo Bill State Park, between the east entrance and Cody. This is a dry site. We have solar, so prefer to pick dry sites, but this state park has electricity and water and has reservable sites.
Baker's Hole FS CG, 2 miles north of West Yellowstone - another dry site, but most are electricity and there are water spigots throughout, as long as you have a "water thief", plus there is a fill up station at the entrance.
An electric site at Baker's Hole
Falls Campground, off Togwotee Pass between Moran Junction and Dubois, Wyoming. Electric Site. Two loops in this campground, one loop is dry, the other has electricity. There's water at the entrance. Last year when we were there the water line was broken so the sites were half price.
Grizzly RV, West Yellowstone - this is late September.
Colter Bay RV - looks tight, but it's not. Huge Class A's make it through here and they place you in size appropriate sites.