Don't know about Arizona or California parts of the trip. Grand Canyon can still be pretty busy, and September sees a big influx of RV visitors from Europe to the Colorado Plateau parks and tourist sites.
From experience, I can tell you along the Old 66 route through New Mexico, Texas, Oklahoma, Kansas, Missouri reservations will not be needed if you are just looking for a place to stop. RV park and campground space will not be a problem after Labor Day weekend, and things don't start closing for the winter until later in October.
I don't know Nebraska, haven't been there yet in a RV, September and October.
RV parks in the Rocky Mountains, if your route takes you to Colorado and maybe NE New Mexico, you might start having a problem with closures for the winter, middle to late September. It slacks off that much. You will find parks open, and spaces in campgrounds, but not everything will be open, the way it is during the busy summer season. A lot of RV park operators go south for the winter, and public campgrounds depend on volunteer hosts who will also flee to a warmer climate.
Get a good RV park and CG guide (I used Trailer Life, guess it is now Good Sam) and check at the middle of the day where you might stop. Pick out some parks in the area, call ahead to see if they are still open. Many listings include the dates if the parks have a season, but dates can change.