No refillable mugs like at WDW.
They have "souvenir" mugs, and they generally come with your first drink. But they aren't refillable like you're thinking.
Now, at the self-service places (White Water Snacks specifically), I have used a souvenir mug for coffee, but when I get up to the place to pay, I pay the exact same amount as I would have paid if I took a throwaway cup.