Thank you for the links. What they say, however, is that ReserveAmerica (the company) is the operating contractor for the Federal Government for the Recreation.gov website. That's how a lot of the government operates; contractors bid to do the actual work.
What these links do not say is that ReserveAmerica and Recreation.gov are physically the same entity or are in any way related other than the fact that ReserveAmerica has the contract to operate the government site.
When a contractor bids to operate a federal facility the proposed operating policies are usually spelled out in the proposal that is submitted to the government and may have been stipulated by the government, in advance in the RFP. If a reservation fee was permitted it most likely was something the original RFP gave the contractor the right to impose.
If the contractor is doing a poor job for the government the best thing to do is to complain the the contracting agency (the government). If you don't like the policies they are using, complain so they can be changed for the next contract. You may not like the way ReserveAmerica runs its own site, but you can potentially have more impact on how the government writes the next RFP to operate Recreation.gov