Will it be hard to get a campsite at Yellowstone in August?

I never sweat over campsites anywhere. However, I am flexible. I would never camp in Yellowstone campgrounds. They are over-developed for me. I did enjoy camping in the off-season in National Forest campgrounds around Yellowstone. If I recall correctly, I was at an empty campground in Gallatin National Forest. I also had a fantastic experience at Grand Teton National Park, including an almost empty campground. I doubt that it fits your needs for facilities, but you can always go dispersed camping if you are rejected at a campground. Almost all of my recent camping experiences are dispersed camping in wilderness areas. I never see bears in my camp when I am dispersed camping. I often meet them in developed villages and campgrounds.