just ask...many of the Disney-owned restaurants allow these options on drinks.
Our experience has been no options at the non-Disney owned restaurants (like Yak & Yeti at AK, Wolfgang Puck at DTD, many of the World Showcase restaurants...Via Napoli, Tutto, San Angel Inn, La Hacienda, Teppan Edo, Les Chefs, and more.) These places tend to lack flexibility on beverages: you're generally allowed Coca-cola products from the soda fountain, regular coffee or regular iced tea. Other options are on the menu, but you'll pay cash for those.