In mid Sept., most of the campgrounds inside yellowstone will be closed.
the best source of rv parks with full hookups is West Yellowstone because there are several rv parks and because the town is right on the Park's border and a decent spot to start your tours of Yellowstone. But call and check on the availability of Fishing Bridge rv campground inside the park. you might get lucky.
For the Tetons, there is Colter Bay RV campground on the northern end of the park
and Gros Ventre park in the south has added some sites with electricity but not sure if you can reserve them ahead of time. Both these campgrounds should be open until late Sept.
I imagine you are not afraid of a little snow, but be advised you might see a snow shower or two at that time of year.