Ethnic restaurants are often good choices but you do have to ask questions. Asian food will usually serve many vegetable and tofu dishes. However there is the possibility of chicken stock, eggs and other animal products in sauces. There are many Asian restaurants now serving tofu substitutes for chicken, fish, etc. Indian food is also a good bet, but you will want to ask about cream, yogurt, butter and ghee in some entrees.
Mexican food has many options such as beans and rice, tortillas, salsa and guacamole.
Be sure to ask about lard in the beans or refried beans. Ethiopian food typically offers a nice range of vegetarian dishes. Italian food is an option, but you will often end up with spaghetti with marinara sauce. Many fresh pastas are made with egg, so ask your waitperson before making your choice. Usually, your best choice is finding a strictly vegetarian or vegan restaurant to please your palate and fill your stomach.