I thought I should put up a list of Korean food or Hanshik (한식) that Muslims in particular can eat at "any restaurant" across the length and breadth of South Korea. This list includes dishes that are served with seafood in it or are simply composed of rice and vegetables or noodles and vegetables. By the way, just to be on the safe side you can tell restaurant manager that: Gogi um-mou-go yeo! (I don't eat meat/ 저는 고기 안 먹어요).
You do not have to look around the halal food restaurants because it is obviously difficult. NOT all neighborhoods offers halal food . It's better that you stick to Korean food that falls within the confines of halal food and there is plenty to choose from. I've short-listed Zen Kimchi's list here and have made some additions as well.
Korea Tourism website has a very informative page on Korean food so check that out here.
Main Dishes being served in a Korean Restaurant
Buddhist Temple Cuisine (entirely vegetarian)
Jeon (Pancake) or Pajeon (Green Onion Pancake)
Yachae Gimbap (Seaweed Rice Rolls)
Bibimbap (Mixed Rice and Vegetables)
O-jingo Bokkeum Bap (Spicy Squid Stew served with Rice)
Boribap (Mixed Barley Rice and Vegetables)
Sundubu Jjigae (Soft Tofu Stew)
Haemul Sundubu Jjigae
Juk (porridges of differnt kinds), Jeonbok Juk (Rice Porridge with Abalone)
Kimchi Jjim (Braised Kimchi with Tofu)
Lotteria’s Shrimp Burger
Lotteria's Squid Burger in spicy flavour
Sae-u Kang (Shrimp Flavored “Fries”)
Doenjang Jjigae (Fermented Bean Paste Stew)
Ddeokbokki (Chewy Rice Cakes in Spicy Sauce)
Mae-eunTang (Spicy Fish Soup)
Nakji Bokkeum (Stir-fried Baby Octopus, I got very sick eating this and then never ate it again.)
Dolsot Yachae Bibimbap (Mixed Rice and Vegetables in a Sizzling Stone Pot)
Dolsot Chamchi Bibimbap (Mixed rice with tuna and vegetables in a stone pot)
Saengseon Gui (Korean-style Grilled Fish served with rice ad side dishes)
Hui Mul (Korean style Sashimi)
Haemultang (Seafood Soup)
Naengmyeon (Chilled Noodles)
Side Dishes (Banchan), Desserts, Snacks or Drinks
Myeolchi Bokkeum (Stir-fried Anchovies)
Odeng/Eomuk (Street-side Fish Noodles)
Hoddeok (Stuffed Street-side Pastries)
Beondegi (Silkworm Larvae)
Golbaengi (Sea Snails)
Jangeo Gui (Grilled Eel)
Sae-u Sogeum Gui (Salt Grilled Shrimp)
Deodeok Root
BindaeDdeok (Mung Bean Pancake)
Ttuekim
Dotorimuk (Acorn Jelly)
Patbingsu (Shaved Ice and Red Bean Treat)
Gyeran Jjim (Steamed Egg)
Corn Ice Cream
Songpyeon (Stuffed Chewy Rice Cakes)
Shikhye (Sweet Rice Punch)
Any product with Green Tea in it
Baek Kimchi (White Cabbage Kimchi)
Mul Kimchi (Water Kimchi)
Oi Sobaegi (Stuffed Cucumber Kimchi)
Ggakdugi (Cubed Radish Kimchi)
Sae-u Jeot (Salted Tiny Shrimp)
Myeongran Jeot (Salted Pollack Roe)
Ssamjang (Mixed Soybean and Pepper Paste)
Yachae Ramyeon (Vegetable Ramen Noodles)
BungeoBbang (Goldfish-shaped Stuffed Pastry)
GeJang (Raw Fermented Crabs)