Yellowstone NP in early Sept

You are going in the right time of the year. If you need more time in Yellowstone, Mammoth and several other park campgrounds are first come-first serve, no reservation needed. Show up on the morning you want to camp, especially in Sept/Oct and you will likely get a site. Once on the site, you can stay up to 30 days. We stayed for 7 days in Mammoth campground in mid Sept/18, I didn't notice anyone get turned away if they arrived by 11:00 am. Mammoth is nice, all pull thru sites, and the very helpful campground staff will get you into the right length of site you need. If you go to Mammoth, don't be alarmed if you pull into a short line up as is normal every morning.

Yeah they are busy in the fall, I saw a clip from a tour guide in GCNP saying that the south rim has more traffic in the fall than in the summer. If people can they avoid the kids summer season.

We've been to Yellowstone NP many times and always go in early September or mid. It's just the best time for us at work. We got in to Bridge Bay once and at that time of year, they block off a portion of the CG leaving just a portion of it open. It's a huge CG, so not sure why they do that. All other CG's there are much smaller, but are full daily. Often, we've stayed outside the park in order to get a site that fits our FW.