For anyone traveling who just has the paper card, you might be interested in this. I just realized that when I log into MyChart, there is a QR code on the main page that documents in detail my vaccine status (first 2 shots and booster). You can access the QR code without even logging in (at least on mine, you can with a quick settings change). If you have an iPhone, you can add it to your regular Wallet and the Health app (I'm sure there is something similar on Androids).
I will be in Germany for a month and I'm hoping that this will be a good backup just in case the white card isn't enough.