The Fishing Bridge campground inside the park is open till Sep 22. We just stayed a week at Colter Bay in Grand Teton just south of Yellowstone. Colter Bay is about 15 miles south of Yellowstone. Colter Bay is also about 48 times more attractive than West Yellowstone. But W Yellowstone does have some great fishing tackle and nice shops. Fishing Bridge CG is pretty darn nice. Check with them if you want to stay in park.
There has been LOTS of rain a couple days ago. We know as we were on a raft trip. Sky was absolutely clear when we were there two days ago. So unless fire started in the last two days it is not an issue.
Finding space. Our experience is that once you pass Labor Day Yellowstone clears out a lot. You should be able to get space. When we walked by Fishing Bridge campground about 5-6 days ago they had a "vacancy" sign out meaning they had space for people without reservations.
Final comment. I love the area just west of West Yellowstone near Henry's, Ennis, Hebgen, and so on. But I can tell you that so do a lot of other people. Entrance into the park is much easier from the south than the west as far as crowds. When you drive from the south you go right to Yellowstone Lake. When you enter from the West there is many times lots of traffic. And the road from the North is much more twisty and hard to drive. We have also stayed in Livingston and driven into the park going south. Being south and going north is much better. If you are lucky enough to tow a Jeep be sure to take the wonderful 4WD road in Grand Teton that is on the banks of the Snake just to the west of it.