Do we have to make an online reservation or do we have to just show up and take a chance that there is a site available

First come first served. No online reservations

I don't believe the BLM campgrounds take reservations. Rio Bravo was not at all busy when we were there a few weeks ago and there are several others (at least one other with electeic) in the valley within a couple miles of Rio Bravo if it happen to be full. You shouldn't have an issue though!