I am not aware of any camping at the Lost Creek Falls trail or trailhead. There is camping nearby in the National Forest.
I did not see camping information; however, I don't think camping is available. The gate for the parking lot has a 6am to 6pm time (if I recall correctly). They lock their gate after hours, and cars left inside have to pay a fine (I think so). Given this, I think campus is not available; but I'm not 100% sure
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