Yellowstone in September

Commercial campground is probably a good idea.

The NPS CGs inside the park start closing in September, and although they've added some cellular capability, I wouldn't rely on getting what you need.

We went to YNP after Labor Day one year and it was wonderful. Great weather (cold at night, though) and moderate crowds especially if you set out early each day.

Stayed at Baker's Hole in Sept a few year's back with a nice 50 amp site over looking the Madison River....it too is a great base for touring.

Grizzly is great as a base camp for Yellowstone, and may very well have sites available after Labor Day with a 2-3 week advance notice. They offer Wifi but have limited time available to campers; if you can bring your own hot spot, especially if you have Verizon service, that should work a little better for your daughter's homework situation.

The park is still crowded - September has been "discovered" over the last decade or so - but it's still less busy than during the peak summer months. Some of the facilities (hotels, restaurants, etc.) in the park will close right after Labor Day, but enough will be open that you can find what you need. Weather is usually good that month; it may freeze overnight, but that's not a problem if you are prepared for it.

Ditto the endorsement for Grizzlys. Fantastic place.

We stayed in West Yellowstone at Grizzlies RV Park, which we recommend. We were there towards the end of Sept and the crowds were not too bad.