You don't have to make reservations on your trip across the Plains. There are plenty of RV parks along the way and if you stop by 4pm you'll get a site.
Since you're on a time schedule make a reservation for inside the national parks. There are also some campgrounds in the parks that don't even accept reservations. For those it would be recommended that you stay nearby the previous night and drive in early morning.
Next to Yellowstone is Grand Teton Nat'l Park so definitely don't pass it by. It's a beauty and altogether different from Yellowstone in looks and things to do. Take the gentle couple-hour float trip down the Snake River in the park and you just might see a moose in the water with you. We did!
Stay in the parks themselves for the best national park experience.
Grand Canyon - Trailer Village has full hookups and the free shuttle stops right there. There are two dry campgrounds which are very nice - Mather and Desert View.
Yellowstone - Fishing Bridge has full hookups. Madison would be good if you don't need hookups.
Grant Teton - Colter Bay RV has full hookups. The two big 300-site each dry campgrounds are Colter Bay (next to the RV park) and Gros Ventre, closer to the town of Jackson. They rarely fill.
Each national park has a great web site which gives things to do, road information, special alerts and campground information. Check them out.
https://www.nps.gov/yell/planyourvisit/campgrounds.htmThis is a good description of all the roads coming into Yellowstone. If you go to Grand Canyon first then Grand Teton; you can continue straight north into Yellowstone from the South Entrance.:
http://www.rvtechmag.com/travel/4_yellowstonebyrv.phphttps://www.nps.gov/grte/index.htm Grand Teton
https://www.nps.gov/grca/index.htm Grand Canyon
