If you need hookups, Colter Bay RV in the park itself is the very best.
If you don't need hookups - either Colter Bay campground (next door to the above RV park) or closer to Jackson - Gros Ventre campground - our favorite. The two campgrounds don't accept reservations but each has 300 sites and rarely fill. Stay outside the park close by and drive to one of the campground in the morning and you'll get a site. We always have with our 40' motorhome. These two campground have a very limited amount of electric sites so ask for one and you might get lucky.
Study the park's website for lots of good information on things to do, road information, camping, maps, etc.
https://www.nps.gov/grte/planyourvisit/maps.htm