I have a card now in my wallet proving the dates of my vaccinations. It is my form of "passport" to enter various establishments.
In Israel, they have what they call the "green passport." Since the fact of vaccination of a person is known to the government, if you have had your two vaccinations, you can download onto your cellphone from a government website a "green passport" allowing you entry into gyms, cinemas, and so forth. Everyone has a cellphone, of course.
I too have such a card, the problem is they could be easily forged--there's no standards nor security on the card. We need a national passport, but I think it's too late now and there could be great resistance to it.