First of all, welcome to our forum; we will try to assist you. Although my diabetes is Type 2, and I am not sure about Celiac, I will give some suggestions.
If you are staying at a Disney Hotel (other than the Home-Away-From-Home or the Grand Floridian) they normally charge $10.00 per day for a refrigerator. I do not have the "Special Reservations" number (which someone else will probably post), but if you contact them and explain the medical necessity they will normally arrange for a refrigerator at no charge. Most off-property hotels will, if you explain the medical necessity, provide a refrigerator for the medications.
Each of the Parks has a First Aid station. You can, during the day, store medications there and do injections there if you want. They also have Sharps Containers for disposal of syringes, etc. If there is need for a break, or other immediate service, they will be able to assist.
A mid-day break away from the Park for nap and/or swimming pool is almost a necessity.
As long as you have priority seating reservations, if you contact a specific restaurant about 24 hours in advance you can make arrangements for almost any special diets.
If you give a little more detail about your child we might be able to make additional suggestions which would have a more direct bearing, such as when you will be going, where you will be staying, and if you will be using Disney (or hotel) transportation or using your own car.
One last (very minor) thing, please turn off your "Caps Lock" key when typing messages, they are generally easier to read in mixed case.