Hi Cheri, I'm glad you asked this, because at my last visit to the Magic Kingdom, I saw a family that had lugged all their family's refillable mugs to the park, only to find out that they were not refillable there. Hopefully if people see your question, they will save themselves that trouble! So, to answer your question again, no they are not refillable at the parks, only the resort you are staying in.
If your family are big soda (or coffee or iced tea) drinkers, at around $14, these mugs will still be a great value. You
could still bring the mugs with you if you'd like to refill them at the water fountains located throughout the parks, but as far as restaurants refilling them, they cannot.
There are separate mugs which may be purchased and refilled at the water parks of Typhoon Lagoon and Blizzard Beach for about $11, and Disney Mom Erin has more information about those
water park refillable mugs here.
I hope this helps with your planning, please visit us again with any other questions! ~Suzannah