September is also a lovely time to be in the park. Usually weather is good, though overnight temps can be at or below freezing by Labor Day (even by late August some years). September has also been "discovered," so if you arrive without reservations, you may have some difficulty finding campsites where you want. That said, you may get lucky, especially toward the middle of the month, just before closing. Try to camp in the park if you can: Fishing Bridge if you need hookups, other CGs (Madison & Bridge Bay your best bets) if you don't. If you want the peace of mind of advance reservations (I always do) and can't get a site inside the park, Grizzly RV Park in West Yellowstone is a very nice CG, should still be open that time of year, and is only a couple blocks from the park boundary.
Be sure to allow AT LEAST 3 full days to see the park, and if possible make sure to spend another day - or route yourself either going or coming via Grand Teton National Park, which is next door (literally - its northern boundary is about 5 miles from YNP's south boundary). One day in Yellowstone is definitely NOT sufficient.