Thought I'd bump this thread to update some of the information.
OKOUTDOOR in downtown Gwangju no longer sells climbing shoes or harnesses but they do have a small selection of quickdraws and belay tools.
To my knowledge Starun in Seo-gu is the best store in Gwangju and can accommodate most of your climbing needs. To get there, you can take a taxi to Sangmu E-MART from which, Starun is about a 10 minute walk (It's next to a fairly big cycling store). I have marked the stores location roughly on this map
and a picture of the shop front here
Whilst in Sangmu, do take a peaceful stroll through the May 18 Memorial Park with its excess of fashionable and enthusiastic Korean power walkers