Sorry - I can't remember if you are flying or driving?
If you are driving, there are dorm-sized portable refrigerators you can bring with you that might be an option - since you would use it only for the medicine, it should be able to hold it at a steady temperature; it's basically the smaller size of the in-room fridge that Disney provides.
There are also devices that are made just for keeping meds at temp:
This model is reasonably priced; I'm not sure what kind of medicine you are trying to keep stable, but if you search
Amazon for "
portable refrigerator medicine", there are several interesting options.