Yes. There are SEVERAL Pit Toilets at La Posa South. Check the map near the entrance/camp host location. At La Posa South there is also a dumpster for trash drop-off, a wastewater Dump Station, and freshwater access. All of this is available to PERMITTED campers.

La Posa South has 10 ADA vault toilets...

There are only a few restrooms up closer to the area where you enter the La Posa South BLM property, where you most likely don't want to stay. To get away from the main entrance road where you can avoid the dust and have some privacy you will need a self contained rig. So if you don't have a self contained rig you will need to stay up by the entrance area close to the ranger station. Certainly not something that we would want to do, but there are some folks camping up there for the facilities.