Inspired by some templates for making cases for Game Boy games from audio cassette cases, I decided to do something like that for Tamagotchis. Just print out a template, cut it out, and place it in the cassette case the same place one of those inserts that comes with audio cassettes go. Of course, if there's no design for the Tamagotchi you want to store this way, you'll want to make a design of your own!
Here is the template:
In addition, I've made an example design for the original Tamagotchi virtual pet from 1997:
You can use cases for old audio cassettes, or you can buy blank cases online at a place like this:
And finally, here's a guide for applying the designs (not mine, just re-uploaded to a more permanent host):