Courtney Boone is correct. The simple answer is "yes, so long as you have an eligible sponsor (active or former military staying with you)."
Only active military or retired military with service connected disabilities and their immediate families can get onto the base. Non military members must be accompanied by their military family on every entry. IDs will be checked. You can call the Charleston AFB Visitor Center and ask them for specifics. All guests must be registered there anyway.
As long as you can get on base you can use the campground. Without a military ID or sponsor that would not be possible.
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