Is camping available on the mountain?

With a permit you can. And you can either drive or hike up. If you hike follow the mounds of rocks that hikers have adding to each time the go or down. That way you can follow the trail up and back in bad weather or snow. Make sure you add you rock to the pile.

You can get permits and all the information you need, across the street from the entrance roadb at the main building.

Some camping available - designated areas. Pay to drive up or choose to hike..park will instruct

I do not find a camp there. But a booking is possible.