As I have now returned from a wonderful stay in Dubrivnik, perhaps I can reply to my own post!
There are no specifically vegetarian restaurants in Dubrovnik, although as a vegetarian it is possible to eat reasonably well. Almost every restaurant has a variant of the "vegetarian platter" on the menu, and this generally consists of a selection of fresh vegetables (for example, aubergines, courgettes, mushrooms, tomatoes, sometimes with rice) which is fine, if a little unimaginitive. It does not seem to have occurred to the chefs that vegetarians need a protein source, and the platters could easily be made more nourishing and sustaining with the addition of a pulse - beans, lentils, chick peas. Omelettes are common, and of course there are Italian restaurants serving veggie pasta dishes. Just a short trip along the coast to the east will take you to Mlini, which actually has a 100% veggie restaurant. Sadly I was not able to sample it, but it looked good.
For a good cheap lunch you cannot go far wrong with the Self Service Restaurant in a side street off the Stranum to the right towards St. Blaise's Church, which has a good range of salads, pulses, potatoes and rice, priced by weight and very good value. Enjoy!