It has already been done. The international agreements are in place from Wikipedia:
"The International Certificate of Vaccination or Prophylaxis (ICVP), also known as the Carte Jaune or Yellow Card, is an official vaccination report created by the World Health Organization (WHO). As a travel document, it is a kind of medical passport that is recognised internationally and may be required for entry to certain countries where there are increased health risks for travellers."
I do know this is for Yellow Fever, others have stated it used to be used to get into and out of Europe and other places for smallpox.
I had mine filled in when I got the shots. The folks at the center said I was far from the only one.
Just build on the existing system. Yellow fever has killed a lot more people than covid, of course it has been around a lot longer. Walter Reed has a famous hospital named for him, due to his work on Yellow Fever.