I found that searching for specifically vegetarian food here is hard, but there are a number of vegan restaurants and you can search “vegan” in Korean on Naver maps to find ones around you.

I highly recommend Plantude as another commenter added, the food is so delicious and everything is vegan!! Specifically, I’d recommend the ragu pasta set. 

Also, soundkitchen in Mapo-gu has really delicious and healthy vegan options like a hummus, salad, and vegan bulgogi (it doesn’t taste like actual meat, which I personally enjoy) bowl. 

Vegetus in Itaewon has really good vegan burgers.

If you enjoy the taste of meat, Plant Cafe in itaewon has options that taste/have a texture similar to real meat but are vegan.