animal.go.kr
The Animal Protection Management System (www.animal.go.kr) has listings of animals caught by Animal Control, and Lost Pet announcements..
List of animals caught and placed in shelters
KARMA
Korea Animal Rescue & Management AC (KARMA) also has listings of animals in shelters.
The now-defunct Animal Rescue Korea website has a few blog posts about how to adopt pets from KARMA.
Paw in Hand
You can view and report lost animals using the Paw in Hand app (Facebook, Naver blog). The app is in Korean.
Misc.
Expats tend to post lost and found animals in the following websites and Facebook groups:
