You would reserve online: Recreation.gov https://www.recreation.gov/search?inventory_type=camping&inventory_type=dayuse
If you want to camp in the park you probably need to make reservations months in advance. There are however many campgrounds near the east and south entrances that you could check out.
The visitor center has a posting today that reservations were needed for camping.
I'm not sure we stayed at one of the lodges near by I think to camp you have to make reservations or it might be first come first serve all I know it's always full
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