Here's the reservation web site:
Facility Details - BURNS RUN WEST, OK - Recreation.gov
The walk up season (no reservations) ends May 14 then reservations begin.
Perhaps there is only a 'iron ranger' taking the payments at this time (put $ in envelope and drop in slot) and that's why no one is answering the phone - no one's there. Public parks sometimes have employees/volunteers do park or campground monitoring by way of driving from park to park in slow times. There could be that no one is stationed there right now. Also, this is in Oklahoma where there's been violent weather recently. Perhaps the park's been affected or closed temporally.
Here's the phone # I come up with: 580-965-4922
Many busy public parks frown on folks arriving early before check-in. The previous party has paid for the site until check-out and then the host usually goes through the site for cleanup. They sometimes don't have room for big RVs to be parking and waiting for the previous party to leave and it's actually rude to keep driving by the site asking, "when are you leaving"? We've volunteered in public parks and have dealt with this issue. If it's not a busy park then there's usually no issue in coming in early and parking somewhere.