International type restaurants that you find in tourist areas, like Italian or Chinese will always have the usual non-meat options. In more typical Dominican establishments you can ask, and they might have delicious vegetable dishes like berenjenas asadas (roast aubergine/eggplant). You can ask for the plato del dia 'sin carne/pollo/pescado' - although the bean stew is usually made with chicken stock cubes. Depends how strict you are.
In Santo Domingo you will find almost every imaginable type of restaurant, including specialist vegetarian ones like Govinda's off Maximo Gomez (next to the supermercado Nacional), Raices, a couple of streets to the south, Ananda in Gazcue and others.
This article has some general, if light-hearted, advice.
Finally, you can do a search in the DR1 message board archives where the topic has come up now and again.
Here are a few results - bear in mind that the further down the list you go the more out of date the information gets.
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Restaurants - Any with vegetarian options?
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