I've made many, many camping trips to Yellowstone and very seldom make reservations ahead of time. As other have stated, some of the FS campgrounds outside the park are outstanding (no hookups) and we prefer to use them, maybe half the time. Inside the park, if we can get there at check out time, we've usually had good luck. Your rig will make a difference. Big ones are harder to find spots for.
When you enter a gate in the morning, ask where there are spaces available. They keep an updated list.
We do it this way because we don't live that far from the park and usually go on the "spur of the moment."