We have this Ice in Germany called Kratzeis which comes in cups and is supposed to be frozen solid and then scraped with a spoon.

This machine turns similar products into slushys. Put the whole thing in your freezer and let it slowly rotate the Ice cup for around half an hour.

As a power source I can recommend just two AA batteries. Get or print yourself a battery holder (there are a few cool designs) and place it ouside of the fridge (the batteries won't work inside).

I used these cheap motors from amazon and wrapped them in toilet paper and tape because they also don't like being in the freezer so much: