They have "souvenir" mugs, and the ones for coffee/cocoa etc generally come with ONE free drink (similar to how Starbucks gives you a drink for free when you buy a travel mug), but you can't refill it for free like at WDW. And even if you go to a self-serve place for coffee, etc, they'll get really worried unless they know you know that you're going to be paying full price for that coffee.
The White Water Snacks lady must have confirmed with me 3 times in different ways that I knew that I would be paying...I had one normal cup (for hubby) and one souvenir mug that I had bought previously, I came up *telling* her that I knew...but she was very nervous about it (I'm assuming she's had some annoyed people regarding souvenir vs refillable).
And the Souvenir Sippers, for soda, can't be refilled at all, as far as I can tell. Only fell for THOSE one time...crazy huge....oh wait, we did it twice...crazy huge Buzz (he leaks!) and Woody souvenir sipper mugs. We don't even drink soda!!!! But DS wanted them and we were still in the process of buying every souvenir they have, LOL.