They will ask you for your sponsor’s name, and service when you check in to make the reservation. So no, general public are not allowed to make a reservation - as of Aug 2023.

Here's what it says under "eligibility": Active Duty, Retired Military, Reservists, National Guards, DoD Civilians, Installation Contractors and those listed in AR215-1, paragraph 7-1 are eligible to camp at the Fort Belvoir Travel Camp. Authorized User Verification: A valid ID card must be presented at check-in indicating proof of meeting the eligibility requirement.