I got vaccinated and I want to attach my vaccination card to my passport. I figure that I can staple or glue it onto one of the pages reserved for visas. I wonder if that counts as modifying it or has unforeseen negative consequences. I have a US passport.
This is a bad idea, for at least two reasons. First, if a US government officer determines that the vaccine card constitutes an "unauthorized change, obliteration, entry or photograph" within the meaning of 22 CFR 51.4(g)(5), the officer could "either take possession of the passport or sends a written notice" to you rendering the passport invalid.
Second, any other country's laws might require its immigration inspectors to refuse to recognize the passport even if the US government holds that the passport is still valid. The last thing you want is trouble traveling to another country because of your vaccination card.
A better approach: use a paperclip, as suggested in the comments, or get a travel wallet to keep both documents. You might also consider a sleeve with a pocket, an envelope, or even a small booklet that you could glue the card into without invalidating the passport, that you could strap to the passport with a rubber band.
Now I realize I haven't actually made a definitive statement about whether gluing the card to the passport is allowed. But the potential consequences of doing it are sufficiently severe, and the likelihood sufficiently low that it is allowed (and that every government officer you meet will agree that it is allowed) that the best course of action is just to assume that it's fobidden.
If you get the COVID vaccine in the US these days, you will be issued a "Vaccination Record Card" that follows the CDC-mandated format. However international travel would most likely require the International Certificate of Vaccination - or at least that's what the requirements seem to be currently. Specifically, that's what Iceland wants to see for non-EU/EEA issued vaccinations:
Certificates from the the World Health Organization (WHO) (the International Certificate of Vaccination or the Carte Jaune/Yellow Card) is also accepted for vaccines the WHO has validated.
Of course these requirements are in a constant state of flux and I won’t be surprised if the CDC certificate will eventually be accepted everywhere but I’d err on the side of caution and get the international version instead. So when you go get your vaccination, fill out the International certificate with your details (a blank version can be purchased on Amazon) and then ask the vaccination workers to stamp it in addition to the CDC paper. If that doesn’t work or you’ve already done your shot, you can always ask your doctor to do it instead.
Afterwards, travel with your passport and your international vaccine certificate. You might still need the CDC card for domestic purposes so I’d keep it safe too but you’re unlikely to need it for international trips.