Am I allowed to glue my vaccine card to my passport?

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.