Grand Canyon (south rim), Yosemite and Zion will be you biggest challenges. The popular campground for Grand Canyon is Trailer Village, inside the park. Yosemite will be the Yosemite Valley sites which include Upper Pines, Lower Pines and a couple of other campsites open part time. The place to be in Zion is Watchman campground that is just inside the park.
I know it's hard to make a hard schedule, but you should book at least the three above, 5-6 months in advance and then book your other stops around them.