We live fairly close to YNP and TGT that it's our favorite place to visit and we plan to go again this September, our favorite and usual time to visit. We've even visited in October seeing a dusting of snow when we woke up! We dry camp a lot as we are self contained and tow with a diesel. Diesel is available as a previous poster stated. Yes, some campgrounds in the park and some outside close mid September, but you will always find one available. Outside of West Yellowstone is Baker's Hole, a FS CG with electric sites and they stay open as long as weather permits. They are right on the Madison River and you can hear elk bugling and wolves howling at night from there. A little further outside of West Yellowstone into Idaho is Henry's Lake State Park and numerous private RV parks too. Red Rock is a little further out, but nice if you prefer that. Within West Yellowstone are others, including Grizzly which is a favorite and late in the year, it's not as busy and they offer discounts, plus free coffee and cocoa every morning! Your view will fit in Madison, our fifth wheel will not, but we have a lot of options, including dry camping or boondocking which is what we prefer. There's also Colter Bay RV Park and Campground on Jackson Lake. Super nice too. We've even stayed off the Togwotee Pass at Falls CG several times (electric sites in one loop) and driven in to the parks. So, there's a lot of options, but Falls can close too depending on the weather since it is a 8000 + in elevation. Last time we were there in 2015, we left early because our weather band warned of a snow storm and we didn't want to be driving down that pass on snow. Check numerous campground links for information:
Campsitephotos.comCampgroundviews.comNational Forest Camping GuideCampendiumYellowstone Park CampingGrand Teton National Park CampingI use all of these to plan our camping. Campendium is my favorite because previous campers post reviews and photos, plus Google Maps displays the exact location for you to view.
Good luck and enjoy.