Is there a way to add a paper vaccination card to the Health app?

Not the paper card. That you could take a photo of and save in Notes or your Photo Album. Check with your state's public health department to see if they consolidate all health records in your state and can provide you with a verifiable link to enter the card into your Health App and Apple Wallet. That is what is done in Illinois. The Illinois Department of Public Health consolidates all health vaccination records and provides a method to add the information to Health and your Apple Wallet.