There is a hotel and restaurant within the park but I'm not sure they sell ice. However, as others have noted it's a short trip into town where you can.

There is a small store we actually never went to and there is also the hotel. Cannot confirm, but a very short drive to the town of Fort Davis that had several stores we went to for groceries.

Three years ago, you could. Not sure about now.

There is a camp/gift store at the Indian Lodge located within the park, but I'm not sure if they have ice. There's a grocery store in the town of Ft Davis (close by) that may have it as well.