From what I've been told, very few Koreans actually camp out a whole night. Usually they bring lots of camping gear to a big mountain area and stay for the day until it gets dark. But those who do camp, usually just pick a spot off the beaten path and set up a tent. Now, this is all heard second-hand and there may be some miscommunication, but it seems like camping isn't necessarily sactioned or illegal. As long as you don't bother anyone, they let you. But they did say absolutely no fires in the mountains.
I heard there are camping stores that let you rent camping gear. It might be good to ask about that.