How do you transfer vaccination card information onto the iPhone?

If you have download the paper proof, and is stored on the device (as a pdf file), the paper proof will be displayed in Apple Files app & Apple Books app.

All PDF files stored on the iPhone are saved in Apple Files app.

See this Apple article: Use the Files app on your iPhone, iPad or iPod touch

https://support.apple.com/en-gb/HT206481

If you do not have a verifiable vaccination record and want to store your card, take a picture of it instead.

https://support.apple.com/en-us/HT212752