Just came back from getting my first shot in Lake County. They gave me a card with today's date and the type of vax I got. They will fill in the info for the second one when I go back in 3 weeks. The space for my name was left blank (for me to fill in) so it doesn't look like a document that would stand up in court or anything.
Some companies (airlines, Bill Gates et al) are working on creating some sort of "digital passport" for the purpose you mention.
kathy