The Mercado Ecol?gico in Mirador Sur sells eggs, and have sold milk, yogurt and cheese in the past, but not at the moment.
There is also a shop called Org?nica (Plaza Catalunya, Piantini) in Santo Domingo that sells quite a lot of imported products and they might do milk and cheese. They definitely have soy milk, soy cheese and soy yogurt.
A lot of dairy production in the DR is small scale and organic by default although without certification, so you might be able to find a place that sells milk, yogurt and cheese that is organic. I've seen a place, I think it is on Ave. Gustavo Mej?a Ricart in El Mill?n, that probably gets its supplies from a small-scale producer.