Try to ask for a vegetarian pörkölt. It's like a stew, Hungarian style, and often comes vegetarian, just ask. Another typical, though decadent, dish is breaded and fried Camembert cheese.
There are also a number of vegetarian restaurants that are very popular in Budapest now, like Govinda Etterem. But I take it you are looking for specifically Hungarian vegetarian dishes? almost 8 years ago
You can try "Lecsó", it's made from pepper, tomato and onion. Sometimes they made it with eggs or bacon, but you ask it without. Also if you eat fish, try the Fishsoup! The others are just snacks and desserts.
here is a collection of the main veggie restaurants of Budapest: http://tinyurl.com/mklfgf3 about 5 years ago