Usually the requirement is only for reservations, but not for stays as such, at least in my experience. If you arrive without a reservation and they have space, you can pay for and stay for a single night as a "walk-in" camper. (Some states have moved to a system where walk-ins, at least after office hours, have to call and make a reservation on the spot via a hotline phone or a cell phone. They still generally permit one to stay for a single night.)
If you can tell online how many sites are unreserved, or get a person on the phone to tell you that, you can usually form a pretty good idea of whether or not a reservation is really necessary to get a site. If half the reservable sites are not reserved a day or week ahead of time, then it's pretty unlikely there will be a sudden mad rush to grab them all within that time. On the other hand, if only one or two are left, you are probably pushing your luck without locking a site in.