One resource that will give you some options is the KOA. They have a trip planner that will show the KOA campgrounds close to your chosen route. While KOA campgrounds might not always be the first choice they are good to have as an option and provide some way to start planning.
Another is the Route 66 rv campground list https://www.theroute-66.com/rvcampgrounds.html
Beyond that its a little harder to find information without pushing in locally to a particular city or section.
Also it will depend on your route. Do you want something fairly direct or do you have a bunch of time. If so you might loop through Southern Utah where there are a number of Amazing National parks
Generally you can disperse camp for free on BLM land and in National Forests although there may be local restrictions.