I feel there may be some ambiguity in the word "passport". It is arguable that we need not only a solution for international travel but for domestic purposes too.
Staying with the passport as an international document it has been suggested here that you need data space in the chip on the passport. I disagree. Think about the ESTA (and for that matter a US visa). That isn't loaded onto the chip. The passport is scanned at the airport and a message is sent to the US to ask if the pasenger is OK to travel. You don't need anything other than details already on the chip. You arrive at US immigration and the scan the passport again and they could quite easily send a message to the UK database "is this person vaccinated?". XML across the internet (plus some security) and it's done.
I've used the US and the UK as an example but the principle should be possible worldwide.