The park itself is governed by the National Park Service. I doubt if they allow any kind of shooting. The dispersed camping area is owned by the Bureau of Land Management. Their rules and regulations would govern the dispersed camping area, not the state of California. You can probably find the information on the BLM website. I don't know the answer, but I did not hear any shooting during the five days and nights that I was staying there.

I don't know about "allow", but I wouldn't recommend it at all. There are people camping all over the area, and there could easily be people camping a short distance away that are hidden from view due to the terrain. Also, it's right on the border of a National Park, so I'm certain law enforcement would respond quickly. I doubt fellow campers would appreciate it either.

I checked and California does allow shooting on BLM land. You should confirm this though as I take no responsibility for misinterpretation. That said, it is quite flat so there is little backstop. Plus there is a pipeline to the north a highway to the south and another road down the middle. I would say that at the extreme east or west end there might be safe opportunity to shoot away from the rest of the campers.

Target shooting is generally allowed on BLM land, but is usually prohibited within 150 yards of any structure or dispersed campground. For the sake of other campers, I would encourage you to find a suitable location much further away and be sure to not leave spent shells, broken bottles and other trash behind.

I am unaware if this is allowed or not. I was there during this time (November) and did not hear anything of the like.

When I camped there were a lot of campers near me. It is pretty flat and has no backstop. I found this information: https://www.blm.gov/programs/recreation/recreation-programs/recreational-shooting/california

As Kaybee said it is relatively flat in this area. To the east of the area there is a campground/ museum/ restaurant and store. That would leave just to the west and I’m not sure if the BLM boundary in that direction.