I haven't tried personally, but I think when the DD Starbucks opens, you may be able to, since that's an actual Starbucks. I think you may not be able to in the Market House or FFP since technically those are the resort selling Starbucks coffee, not an actual Starbucks franchise location. I could be wrong though.
I learned recently Starbucks at WDW are able take Starbucks gift cards as payment. And I experienced it myself just last weekend using my gift card to pay for a drink at the Magic Kingdom location (and I earned a star from the purchase). Not sure if the ones at Disneyland do as well, but I'm betting they do.