It's very possible, particularly at the bigger crags (Ganhyeon, Seonunsan, Seoraksan). On the weekends, most Korean gyms head out to climb outside, so if you're staying somewhere specific, you could probably negotiate to join a gym for a month. If you want to travel and see a lot of places, maybe consider camping. Shoot me a message and I'll be happy to give specific advice, etc.
"If you can't do something well, you might as well learn to enjoy doing it poorly."
-- from a de-motivational poster, but I find it oddly liberating