A lot of people have reported that they asked Housekeeping for a sharps container when they checked in and got it delivered promptly.
If you want to check beforehand, you can call WDW RESORTS/SPECIAL RESERVATIONS
(407) 939-7807 [voice]
(407) 939-7670 [TTY]
They really don't want to take a chance that people might not have a place to dispose of syringes, so it's in their best interest to provide one for you.
The word on getting a refrigerator free for medical reasons is mixed. Some people have reported getting one for free, but more people have reported they were told it would be $10 a day and if they didn't want to pay that, they could store their medication behind the front desk.