Wow! That's going to be a challenge to do in a month.
Yellowstone isn't too big a problem. They have some first come first serve campgrounds. ex. Norris Go to Yellowstone!
Like Al-n-Sue said weekends will be the biggest challenge. We are traveling in the west now and haven't made reservations in advance...so far! Weekends are the biggest challenge for us.
California has been the toughest state, especially Yosemite.(Death Valley will be easy in July or August!
) We're on the Oregon Coast now. It's easy. There are state parks, county parks, and independents out the wazoo here. Utah should be okay as you can always camp on BLM land which is plentiful there.
The references that were suggested should be good along with AAA camp guides. Also if you have time research the areas where you plan to go looking for public campgrounds that are fully or partially first come first serve.
Have fun!
trex