Korea is all about meat. Most Korean restaurants serve meat, if not, side dishes containing dairy products. Not many Koreans are vegetarian or vegan, so its really difficult to find one. In many popular places, all you see are different Korean bbq restaurants, no proper vegan restaurant in sight.
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Fortunately, in the between Wangsimni and Sangwangsimni station, there is a vegan buffet which was recommended by some people I know. Hyoso-eh (효소애), has a vegan buffet that costs 13,000 won per adult.
If you go out of exit 1 in Wangsimni station, and keep going, you'll find the restaurant!
For vegans and those who want to try, visit the website for more information!
http://wangsimni.com/detail.php?number=91&category=&PHPSESSID=2414f11a6f5a678428f048b99f666f20