You cannot drive directly through the park to any of the attractions from this campground (you need to leave the park and re-enter) but it is not a far drive and has the added bonus that it is close to all the amenities in Yucca Valley and Joshua Tree, including a gas station and the Walmart.
We bought a threaded butane/propane camp stove canister (the red MSR ones) at Nomad Ventures since Walmart only had the green Coleman ones stocked. Nomad Ventures is right across the street from Coyote Corner where you can have a shower for $4 for 7 minutes. Both are located a short drive from Black Rock Campground.
The campsites themselves are nice and have some trees / shrubs. We went in the off-season and found the facilities to be clean. There is potable water.