Try to stay in Grand Teton Nat'l Park at Colter Bay RV. It's the best place. The RV parks in Jackson are the pits and are very expensive. Headwaters RV is also nearby but Colter Bay is much, much better.
Do you need full hookups? In the national park are two great dry campgrounds and each has 300 non-reservable sites. We've been there with our 40' MH all seasons and always go a spot. Gros Ventre is fairly close to Jackson and it's our favorite.
https://www.nps.gov/grte/planyourvisit/camping.htm