there is a federal campground just below the dam at Palisades Reservoir just west of Alpine, WY. It isn't fancy but has elec, water and dump.
Henry's Lake State Park (Idaho) might not be open so call ahead. It is 15 miles south of West Yellowstone and is a very nice state park with hookups, etc.
Or you can go on to Jackson from Rock Springs and stay out at Gros Ventre campground in the Tetons national park. water and dump station. close to town.
Or, you can take 191 north of Rock Springs and there is a fed campground about 20 miles northwest of Pinedale, water and dump.