My veggie partner has suggested I do this tip, as negotiating the food in Dubrovnik was a little bit challenging initially. It still seems to be very much a meat and two veg cuisine, and veggie options often seem to be nothing more than the vegetable garnish. Otherwise the options are often based on eggs or cheese, making it unsuitable for vegans. Most restaurants offer a vegetarian plate. My partner had this the first night and never went back - it was a dull mishmash of random cooked vegetables plonked unceremoniously on a plate and completely unexceptional. Maybe other places were more creative, but he didn't want to risk it.
Despite this rocky start, he ended up finding enough nice things to keep him happy. His favourites were:
Salads - he rarely eats salad in the UK but the raw vegetable were so much better there (with the exception of cucumber, which was often bitter) that he really enjoyed them
Pizza - not the most original choice, but dependable and, as many other people have pointed out, very good in Dubrovnik too. We ate at the Pizzeria Barracuda where the service was excellent and they had a huge number of veggie choices)
Pasta - especially from Restaurant Penatur, but remember to ask them to leave out the meat.
Risotto - I was surprised more places didn't offer a mushroom risotto, but when they did, it was invariably delicious.
Rikula - see my separate tip under restaurants for more details of this restaurant, which turned out to be the preferred choice for me as a confirmed omnivore as well as my veggie partner.