https://health.wyo.gov/wp-content/uploads/2020/10/2020-COVID-19-shot-card-2b.pdf
Print the card above, 2 sided, on some heavy card stock.
Find a Facebook post from people in your area who have had the vaccine and posted a pic of their vaccine card. Or ask to see the card of someone you know who got it, and take note of the vaccine info.
Copy the vaccine info from their card to the one you printed. Some people are getting a stick on label, some are getting the info hand written, some are coming with the info pre-printed on the card.
BINGO, you now have had the COVID Vaccine!
Take note of where and how that person got their vaccine so you can use that story if you are questioned.
Currently trying to determine if a "National Database" is being created, and if so how it may come into play down the road.
At least with this, you will be able to get around any simple local restrictions.
Once the word gets out these cards are being made, I am sure that they will try to go with something more formal that is connected to your Drivers License or Passport.
If it comes to that, things are going to be on a whole different level!