I did a rather deep dive to find out about how to add cards to my Apple Wallet. I've just cleaned out my Apple Wallet and deleted several expired rewards cards, gift cards, etc. I wanted to find out to add new cards, and this article helped. It's really up to each vendor's app.
Initially, I started the search because I would like to add my COVID-19 Vaccination card to my Wallet. I still haven't figured it out, so if anyone has an idea, please share!