I know for a fact that La Brea Bakery at DTD, has Soy milk for all coffee drinks and I think they also had Chai, but I am not 100% sure of that. Compass Bookstore at the other end of DTD probably offers both as well.
I have never ordered black coffee at DLR since there coffee is terrible so I am unsure about Soymilk for that. But I can tell you I had some sort of flavored Tea Latte thing at DL on Main Street and it was the most awful thing I have ever tasted, DH agreed.