I think you get the mugs and refills at Captain Cook's. They have restaurant's at the Polynesian, but, I believe Captain's is kind of their "food court" fast food type place.
Captain Cook's is indeed where you get your mugs and refills. The menu is somewhat comparable to what you find at the fast-food spots in other deluxe resorts. The variety is, of course far less than you would find in an All-Star food court, but here you have those other two resort reataurants (Ohana and Kona Cafe) plus room service in addition to Captain Cook's.
We rented a fridge from the Poly and stocked it with milk, juice etc. from Captain Cooks. We also bought cereal and fruit. Each morning we had a nice relaxing breakfast before we headed to the park. More nutritious than fast food and very convenient. DS5 would eat breakfast while DH and I got showered and dressed.
I just returned from the Poly a few days ago and agree with Kathy's post above that those are most of the items available at Captain Cook's. I . Also, you can get some of the packaged items (such as chips, drinks, snacks) cheaper upstairs at Samoa Snacks in the Poly.
The refillable mugs have now gone up to $10.99 (before tax, I think). Captain Cook's also has a microwave you can use, and they sell microwave popcorn.
The hot food is burgers, fries, chicken sandwich, chicken strips, and individual pizzas, plus some hot breakfast food. I only ate the burgers, and they are between $5.00 and $6.00, but that includes the fries.