There are two campgrounds in Grand Teton Nat'l Park that don't even accept reservations and they each have 300 sites. It would be no electric but they do have a dump station and water fill. They are Gros Ventre and Colter Bay.
Grand Teton CampgroundsThey rarely fill and if you get there by 1pm you'll definitely get a site.
Yellowstone also has many campgrounds that don't accept reservations. With the small size of your RV you could use any one of them. We've easily gotten into Mammoth with our 40' MH but we arrived early morning.
Yellowstone CampgroundsIf you're uneasy about doing that I would say get a reservation for Yellowstone but wing it for the Tetons and you'll be fine.
If your trip is for early June that's even better. Have a good one!