Harder than it looks, for one specific reason: broth. Knowing where it hides makes it manageable.
Most Korean soups, stews and even many vegetable side dishes are built on anchovy or beef stock, and kimchi is usually made with salted shrimp or fish sauce. A dish can be entirely vegetables on the plate and still not be vegetarian. This is the single thing to check, and it is not obvious from the menu.
Bibimbap ordered without meat and egg (and with the gochujang checked), japchae if made without beef, kimbap in its vegetable version, sundubu ordered as the vegetable one, buchimgae (vegetable pancake), and the temple-food restaurants below. Asking is still worthwhile — 'usually' is doing work in that sentence.
Korean Buddhist temple cuisine (사찰음식, sachal eumsik) is vegan by definition, and additionally avoids the five pungent vegetables including garlic and onion. There are dedicated temple-food restaurants in Seoul, particularly around Insadong and Jogyesa Temple. This is the most reliable option in the country and also some of the best cooking.
'Jeoneun chaesikjuuija-yeyo' (저는 채식주의자예요) — 'I am vegetarian'. More useful in practice: 'gogi-wa saengseon ppaejuseyo' (고기와 생선 빼주세요), 'no meat and no fish'. For broth specifically: 'gogi yuksu-do an dwaeyo' (고기 육수도 안 돼요), 'meat stock is also not okay'. Showing this written down works better than saying it.