If you ask the owner most tavernas have some form of vegetarian food ( unless your a vegan or have specific requirements). Many greeks only eat meat and fish several times a week rather than every day so they do have a wide range of vegetarian dishes. Often the stuffed tomatoes are veggie and you can always make up your main meal from the starters, fried courgettes,aubergines and peppers. Gigantes, fasolada and a variety of salads also make an enjoyable meals,especially with platefuls of chips. Then theres mushrooms a la grec, saganaki and ask about the 'horta' which is seasonal green vegetables.
I've got to go now cos i'm hungry!!!!!!!!!!
Good luck and don't be afraid to ask