I work at an ice cream store that also is a gas station and we sell pints and half gallons of our ice cream in ice cream cases near the drink coolers too. How would I say “fill up” in the sense of filling up the case with the ice cream pints that are missing?
In this context, 補充 is probably the most natural choice. For example you can say ケースにアイスクリームを補充した, アイスクリームの補充を忘れてしまった and so on. If you really need to say the case becomes full of ice cream, you can say something like 補充していっぱいにする, but it's usually unnecessary.