Proof of COVID-19 vaccination

I wouldn’t staple anything to the passport. Just keep the card with your passport. I don’t know if it’s legal to staple it to a passport page, but there’s a good chance you’ll need your card, in the U.S. or overseas, without the passport, e.g., when you get a booster or have to upload it to an app prior to overseas travel. We have a passport holder similar to a wallet with space for the passport and slots that look like credit card slots where we have the card.

In another thread, https://community.ricksteves.com/travel-forum/packing/gifts-for-travelers, there was a discussion of passport holders with a clear plastic window in which you could put your CDC card.

Is it stapled to one of the back pages, or secured in some other way?

I used a paper clip.

I heard Rick say on the radio that he keeps his CDC vaccination card in his passport.

I suspect it is just put in like a bookmark

Affixing it with anything more permanent than a paperclip would be ill-advised.