I wouldn't worry about it there. There aren't any animal warnings in the campground. You'll probably have more trouble with the kangaroo rats.
good idea, by the way - amazing scenery.
there is no water in the park, including the campsites, you have to bring it all. you could probably bum some off the climbers, or get someone to carry some up for you. if you are entering from the town of Joshua Tree that should be no problem, ask people at Nomad or Coyote Corner or Crossroads Cafe. The water at the back of Coyote Corner is good and free, there's showers there too for a few $.
ALso, not sure what the regulations are there about camping outside the campground - you might need a backcountry permit, and to be a certain distance from the road.