I'm going to go with a yes, I've never been to a place in the area that did not allow them, Standard 6 foot lead under control. As a past member has said this is not a campground it's open desert. Borrego Springs is not far away for some creature comforts and touristy spots that are not tourist traps.
FWC just confirmed on the FB page yes dogs are OK with Leash.
Borrego Springs is where all the metal art is? Great place to visit.
Also that means Julian isn't too far away either I think.