No idea what a vaccine card is. When vaccinated you receive a paper or emailed pdf receipt. Similar receipt as you would get with a flu shot but the COVID one is much, much more detailed. It's not used as proof of vaccination except to show your doctor if need be. Businesses aren't going to accept it if that's what people are thinking. That's a different issue to be figured out.
Thinking back on some of the doubts in this thread about MPAs having received COVID vaccinations, perhaps that girl you think is in her 30s is actually in her 50s or 60s