We live in Utah and go to Yellowstone a lot. Sometimes the campgrounds are filled up is September, but mostly not a problem finding a place. The big thing you can do to improve your chances of getting a spot - and a good spot - is the time of day you pull in. Too early and other folks haven't pulled out, and later in the day they fill up again. Getting there from 10 to 2 is good timing if you can make it work.
While the crowds begin to drop off after Labor Day - it's also at that time that they begin closing some campgrounds, and shutting down some of the 'loops' in the campgrounds that are still open.
But even in the summer when crowds are at the peak we've almost always found a spot if we get there in that middle of the day time frame
2005 16' CCD
2005 Dodge Ram 2500 Quad Cab 4WD