Sushi Ya!

If you're a fan of Japanese food but are vegetarian, you can often find dishes to suit you. Maki can be made with avocado, carrot, mango, cucumber. There are rice dishes, and salads with algae are often more interesting and tasty than the usual dull everyday green salad.
Sushi Ya! offers a very reasonable lunchtime menu. Ask for the menu Tendon without the seafood ("sin marisco" or "solo verduras") and you will get a very good main course to accompany the miso soup/gazpacho starter and udon (cold Japanese noodles)/ salad second.
Add this to very quick, friendly service and you can share a pleasant centre-of-town midday menu with any of your fish or meat-eating friends.

La Veronica Pizza Restaurant

Being a vegetarian and an avid pizza fan, I have visited this place many times. The atmosphere is that of nice dimmed lighting (inside), house music and friendly staff. They have an outside terrace in Plaza George Orwell and the food is very tasty - lots of interesting, well-presented salads and some unusual and very innovative pizzas.
My last two visits however, have left me rather disappointed. Reason being the service. I'm not sure how many staff they have, but it seems that they are lacking somewhat in the kitchen department. The last time I went we waited an hour for our pizzas. Then this time, we had to wait 3/4 of an hour after our salads and I think it's fair to say that many other customers were looking around with that "Where the hell is my food" look.
All good things are worth waiting for, but come on, we all have our limits - employing another chef or two would certainly make the experience far better and I don't think they have cash-flow problems as the place is packed practically every night.