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All of the modern ships, I believe, have a refrigerator in every grade of cabin, just check when you book and you can make the booking conditional on that fact. I would not trust anyone else to keep them totally and permanently at the required temperature. There should not be a problem though and if you need it just ask the cabin steward for a sharps box and he will provide that too. ..... Neil.
We cruised on Arcadia to the Baltic capitals one of her maiden cruises in 2005 and had refridgerators in the cabin , I beleive all there cabins have them.
Ocean Village 1 and 2 have fridges in all cabin, but take a fridge thermometor with you ( you can buy them cheaply ) to be ensure that the fridge is running at your required themperature. As Neil says ask for a sharps box.
Not all ships have fridges in the cabins. However most (all?) have medical centres and a Doc and Nurse. They will be able to store any medicine for you, if required.