Most of the vegetarian restaurants in Reykjavik close early, so I decided to try Italia Restaurant (which is open until 11:30 on week nights). I was pleasantly surprised by the food and the service. There weren't many vegan-sounding items on the menu, but the staff was happy to prepare the penne al arrabiata sans-bacon or the pasta di Napoli using egg-free noodles. The rosemary flatbread was vegan without having to make any special requests.
Vegetarians have even more choices, including the ravioli con funghi and a vegetable risotto dish (not to mention various pizzas).
Note for vegans: the pasta dishes automatically come with cheese-covered Italian bread, so make sure to request that the cheese be left off. The Italian bread (without the cheese) is delicious, homemade and dairy-free!