Yes, they do have refillable mugs that can be filled at Captain Cooks (the counter-service eatery).
The Polynesian has a few little shops scattered throughout the Great Ceremonial House. One has snacks, beverages and Disney souveniers, one has clothes and toys, and one has Polynesian type items (coconut heads, little ceramic tikis, etc). I wish they had just one large store like they have in many of the other resorts, but after some searching, your friend should find what he/she needs.