We were in Yellowstone for much of July. We got a 4-night reservation at Grant Village and a 3-night reservation at Bridge Bay only a few days ahead. Then we moved to Mammoth (first come only) for 10 nights. At the end of July Mammoth was filling in the mid/late afternoon.
All the visitors centers have the time the campgrounds filled the day before. We almost always use the internet when booking reservations. However, the Xanterra phone operators have more up-to-date info than is on the web. They can sometimes find a site when the web shows no vacancies.