Lanta's local food is southern Thai and seafood-led: massaman curry, khanom chin (rice noodles in a fish-based curry), grilled fish and shellfish, and chicken biryani that reflects the island's Muslim communities. Beach restaurants display the day's catch on ice. This is a hard island for strict vegetarians: fish sauce, shrimp paste and shared oil are common even in plain-looking dishes. The Thai phrase for vegetarian food is "jay", and you can ask for "no fish sauce" and "no egg", but kitchens vary.
We could not verify any pure vegetarian, Jain or vegan restaurant on Koh Lanta from sources we could open; only K. Himalaya on Long Beach carries a tag, as an Indian restaurant, with no vegetarian claim. Wikivoyage's island guide notes that one resort serves Indian and Thai food, and describes at least one cafe as vegan and vegetarian friendly, but neither is confirmed by the venue itself, so treat them as leads to phone, not promises. Vegetarian, vegan and pescatarian classes are offered at the Time for Lime cooking school, which is a good way to learn what to ask for.
The practical answer is to book a resort that will plan meals with you. Ask in writing, before you arrive, for a vegetarian or no-onion-no-garlic menu, and carry packed snacks on boat days, where lunch is usually seafood. Fruit, fresh coconut, sticky rice, corn and plain fried rice made with no egg are the safest everyday choices.