Wyoming Camprgrounds

This websites shows government campgrounds. We use many of this type.
US campgounds

May is sometimes too early to get into Yellowstone because of late snows. We've been there in late June and had to deal with lots of snow still on the ground at higher elevations. West Yellowstone has several good campgrounds right near the west entrance if you can get through that early. Best of luck.

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.

A Google search for Alpine camping found;

Palisades Cabins & RV

Greys River Cove RV

No experience with either of these.

I also recall passing this campground at the base of the Palisades reservoir dam on a past trip - Riverside Park Campground.

Closer to West Yellowstone is Henrys lake State Park.

Lots of RV parks in West Yellowstone.

In 2014 we spent 4-5 nights at Bakers Hole Campground - just 3 miles north of West Yellowstone and right on the Madison River. One loop with electric.

I've stayed in Boulder and Daniel from Rock Springs to Jackson.
They are ok.

We stayed at Red Rock RV park in Idaho (near west entrance) and found it very nice.