Don't forget a stay in Teton Nat'l Park which is attached to Yellowstone. Both are huge so don't try to do both from one location. If coming from CA you might want to enter Teton from the south. If you need full hookups, stay at Colter Bay RV in the park itself. For no hookups, either Gros Ventre campground in the park - 300 sites - no reservations or Colter Bay campground which is right next door to Colter Bay RV. The campground also has 300 sites - no reservations. For both dry campgrounds I've never heard of anyone not getting a site. Gros Ventre is our favorite.
Then drive north through the Tetons and enter Yellowstone. If you've tried for Fishing Bridge without luck, that's the only place with hookups in the park. However, there are plenty of non-hookup places in the park - some no reservations; some accept reservations.
Here's a web site for the campgrounds:
Yellowstone's CampgroundsIf staying outside the park, West Yellowstone would be the most convenient.
Here's a campground location site in the park:
Yellowstone Campground LocationsHave a good trip!