South of Wall, SD is a really nice boondocking spot... Free RV Camping - Wall, South Dakota
Public National Grasslands.
I'd drive by Mt Rushmore. Crazy Horse monument is a work in progress... don't expect to see a lot more than you see from the highway entrance.
The commercial campground at Devil's tower is pretty nice, and the campground inside the monument has nice views.
Grand Teton National Park has the Gros Ventre campground on the south end of the park. We've never seen this campground full. No hookups but dump and fresh water station. Yellowstone and Grand Teton are in bear country. There are length restrictions in some of the National Park campgrounds... not so for Gros Ventre campground. (Pronounced Grow Vaunt)
National Forest campgrounds are available all over the west. Speaking of National Forests, did you know that Nebraska has a National Forest? Click here for more info... Nebraska National Forests and Grasslands- Home/
When we travel, we prefered state park campgrounds to RV parks whenever possible. Private parks are pretty convenient, but some of them really pack folks in tight and their rates are plenty high. Public state and federal campgrounds may offer power and water, and dump facilities a short distance from your campsite. For example in the Denver CO area... Chatfield Home Page | Colorado State Parks
Don't forget state and federal recreation areas... lots of possibilities.
Bureau of Land Management might have the type of camping you are looking for if you are into remote locations. Many BLM sites are really remote, while others in the Colorado mountains are close to tourist areas... BLM Colorado | Recreation | Camping
Across Nebraska on just about any highway other than the I-80, town parks offer free or donation camping with power and water, and a dump station nearby too... RV Camping - Cambridge, Nebraska
is just one example.
Theodore Roosevelt National Park in North Dakota is off the beaten path a bit, but a great place to avoid crowds and see some fantastic badlands most folks never go to... http://www.nps.gov/thro/
This web site has information and links to most public and private campgrounds where ever you might travel... www.rv-camping.org
The time you plan to travel is ahead of peak season in Wyoming. You will likely see snow in Yellowstone along the road at the least, and some roads may still be closed. There are a lot of twisting mountain roads into and out of Yellowstone and Grand Teton.
That should keep you busy researching for a day or so...