If you can't get a reservation for Fishing Bridge inside Yellowstone, there are a number of private RV Parks in West Yellowstone. The one that gets the most mentions on the forum is
Grizzly RV Park, which is just a few blocks from the west entrance station. We have stayed there a couple of times.
For Grand Teton,
Colter Bay RV Park is inside the Park and is the only campground with FHU. If you prefer to stay in the town of Jackson (Jackson Hole is actually the name of the valley), try the
Virginian RV Park, which is in town. If you can do without hookups, there is a Park Service campground at Colter Bay and the Gros Ventre Park Service campground closer to Jackson. We always stay at Gros Ventre for the Tetons and will be there again in early June.