No food court but a reasonably-priced "fast-food type" grill downstairs in the main building near the pool. It's called Captain Cook's and it is opened til 11 or midnight. They have cereal, pastry, fruit, burgers, chicken sandwiches, candy. Also Mickey Head Ice Cream Bars! Not sure if you can reuse an old mug. Debate rages here; you can certainly try.
Also at the Poly is 'Ohana, an all-you-can-eat Polynesian dinner buffet (also open at breakfast I believe), and Kona Cafe, which serves all 3 meals. They have great coffee and dessert. We had an amazing meal here at Thanksgiving. Something for everyone and those desserts are to die for... Also wonderful homemade bread...
You will not go hungry at this resort.