No need for campground reservations in May, but by mid-June you might want to reserve.
Here's a tip: our absolute favorite Black Hills campground - and we've camped in most of them - is the little Bismarck Lake CG inside of (but not part of) Custer State Park. It's actually a USFS campground, but you have to enter the state park to get to it. Why is it a favorite?
1 - It's quiet - no motors allowed on the lake (most of the BH campgrounds are plagued with jetskis and motorboats).
2 - It's just a few minutes away from the Custer State Park Wildlife Loop, a wonderful place to visit at dawn and dusk for good views of bison, prairie dogs, antelope, foxes, mule deer, elk, mountain sheep and feral donkeys - and wonderful in May when the newborn animals are nursing.
3 - It's also just a few minutes outside the town of Custer, very convenient for shopping and dining.
4 - The south end of the Black Hills is less touristy and crowded, yet convenient to Mt. Rushmore, the Iron Mt. Road and Needles Highway, Crazy Horse Mtn., and an easy drive to Hot Springs and the superb Mammoth archaeological site.
We camp at Bismarck Lake almost every year, usually in late May.