It'll be kept in a self-storage place over the winter, and will not be in a climate-controlled environment. What should I do? Is there a way to purge the water from them? My fridge does have an icemaker, if that matters.

BTT

IM DrFrige and see if he can help you out.

I am not sure on purging the water out, but be sure to somehow prop the door/doors open on the fridge so it wont start to smell/mold on you.

I've never had problems with small amounts of water left in washing machines and I think your fridge should be fine. One problem I have had with storage is mice. I have had them crawl into the insulation and nest, only to die. Bring that into the house and It'll smell real good when it warms up. I cover as many holes with cardboard and tape, and put bait in the room-in the corners.

Scrub that fridge hard before you store it.  Go over it with a diluted bleach mixture as well for safety.
Make sure you drain the washer and completely!  One sub-zero day and a little bit of ice can seriously fuck you over.

If they will be in storage for a long time, the rubber seals can degrade, so don't be surprised if you need to fix a few leaks when you pull them back out.

When it comes down to it, if you're talking really long term, and you don't really love these appliances, you might just be better off selling them and rebuying when you need them again.