You can buy a cheap "barbeque pit" almost anywhere for about W10,000, and these are increasingly becoming more popular at campsites. If you are in a National, Provincial or City Park-- there is usually a campground or check with the officials about camping. At many if not most parks, camping is limited to these campgrounds (which are cheap, but sometimes crowded). Wolchulsan has a nice grounds for non-auto camping as does Gyeryeongsan) and you can hike to climbing. Many crags offer camping (official or set-up and maintained by climbers) near-by, like Jukjang, Sinban, Chuntaesan, Baekseok, Yongseo, Halmaebawi, Hagampo (there's a big solar powered campground with amenities or some beach camping about a kilometer or 2 south with bathrooms) Munsusan, Daeryuk (absolutely no fires here!!) Ganhyeon (fires must be in a pit here, but they will lend you one with a deposit that is refunded upon return). . . many Korean websites
(and the camping magazines) list popular (and a few less-popular) destinations. Be aware that summer is camping season here-- despite the rain and mosquitos!-- so places can get crowded and you might have to book in advance!