Some information on national and State Park camping reservations.
You will have a very difficult time finding reservations for this summer now. We have a similar trip planned for the fall. National Park Campground reservations open 6 months out to your arrival day. Utah State Parks reservation window is 4 months out from arrival date.
I just tried to make reservations at Zion the second the window opened with no luck last Friday. It is much more difficult to find sites with a 30+ foot RV and with construction occurring in many parks.
Try your luck and if everything is booked, keep checking back. People cancel all the time.