First come first served state and national parks

Mammoth Campground is open all year. It is at a lower elevation than most of the CG's in Yellowstone. Not as much snow, but could be cold in April.
You would most likely have to enter from the North Entrance, since that is the only entrance open all year. Most likely the road thru Lamar Valley will be the only road you can travel.

I think April is too EARLY.