Hello. I am a frequent visitor to WDW resorts and park and I also wear a cooling vest and belt.
First Aid will not be able to accommodate the packs. They have a small hotel fridge with an even smaller freezer compartment. Lesson learned.
I have tried Bell Services and they are hit and miss as well. Some hotels are great, some are nervous due to liability. Just make sure you have ziplock bags and your name and room clearly printed on them.
Resturants in no way will take them, even if they are in sealed bags due to contamination issues.
I would suggest (what we have done) is contact the hotel you are staying at direct. Ask them due to medical issues to arrange for a "dessert freezer". It has drawers and is only to freeze. This will take up a corner of your room, as it will NOT fit in the spot of a reg hotel freezer, its a bit larger. There is NO refrigeration just freezing capabilities. They don't have to provide this for you (you understand) but working with a hotel manager via calling/and putting the request in writing via fax will help. ( you can also tell them that CM and hotel staff refer to them as a "dessert freezer" they use for catering, also conventions and weddings. And it's white with drawers)
Then, for the parks I would suggest getting this soft sided cooler
http://www.polarproducts.com/polarshop/m/viewPrd.asp?idproduct=1920&idcategory=0 from Polar Products. I use it through the parks and keep my extra set of cooling packs in there to change out.
I hope this helps some, please feel free to contacted if you need any further information. Blessings- Cyn