I have been an insulin dependent diabetic for 28 years, and on an insulin pump for the last 12 years. I have been to Disney over 20 times. I never had a refrigerator for my insulin in my room before joining the Disney Vacation Club (as even the studio has a refrigerator). So if for some reason you cannot get a refrigerator, do not panic. As mentioned previously, the bottle you are currently using does not have to be refrigerated. For example, the insulin in my pump is not refrigerated, it is in my pump. Any extra unopened bottles you take can be kept in a cooler. Keep them in a ziplock bag to keep the boxes dry.
If a diabetic is in your party, you will/should have a cooler anyway for snacks, so it's probably not something "extra" to bring. As far as needing to be near the food court, if that doesn't work out, just be sure to have plenty of snacks in the room AND on you at all times for unexpected low blood sugars. Excellent portable snacks are raisins, peanut butter cracker packs, hard candy, granola, granola bars. In my opinion, if your sugar is dropping, a quick snack will get your sugar up faster. By the time you get to the food court, even if you are close, wait in line and get the food, the sugar will have dropped so much, it will take more food than just a snack to get it back up.
One other tip I would add - check your blood sugar OFTEN while at Disney, at least every 90 minutes. Even though I've been diabetic for nearly 30 years, I still find my blood sugar hard to control at Disney. I think it's a combination of the excitement, the fact that I'm getting more exercise than I usually do, and eating more snacks than I usually do.
Have a great trip!