Go to the website. The reservation system is very user unfriendly but stick with it and you'll get a reservation. GR has one loop that is First Come, First Serve.
Our first week there we camped in the first come, first serve loop. If I remember correctly that’s loop C. We then made reservations in the office for a week in February and another in March. Call the campground to make your reservation.
If I remember right, we just registered at the campground (in September), but Pima County has a site for registering. Here is the link: https://webcms.pima.gov/cms/one.aspx?portalId=169
https://apm.activecommunities.com/nrpr/Home
You will have to create an account and then pick your spot on the map. The most user friendly but will work for you.