I agree with WillowBelle, there probably isn't much you can do other than frozen water bottles and those probably won't last the whole week. I'd just bring different snacks. How were you going to get them to Disney?
If you're packing them in a cooler full of ice, maybe you could just bring the cooler in to your room and keep refilling it with ice from the ice machines?
Cheese sticks, fruit and yogurt don't need to be frozen though.....so if you're just bringing those I think you'll be fine putting them in the fridge. Bring a little fridge thermometer if you're concerned about the temp of the fridge part being cold enough.