When we were there two yers ago, you could not. The only place you could refill your mug was Captain Cooks. However, when you order coffee or soft drinks you can get free refills. We are returning to the Polynesian in November and can't wait. I talked hubby into upgrading from CBR!
No you cannot use the mug at Kona's. That would be nice though but the coffee is four times more expensive then the stuff at Captain Cooks. And no the coffee at Cooks is crap and nowhere near as good as Kona's either. So you will have to put out the real money if coffee is important to you.