My App Store will not let me add any apps. Not even free apps

Then there's an issue with the card as far as the phone is concerned. It could be as simple as the CVV number needs to be reentered on the phone. But you should contact iTunes Support for assistance with this.

Contact iTunes Support --> iTunes - Official Apple Support

Also see this link about managing payment methods on iPhone --> Change, add, or remove Apple ID payment methods - Apple Support

For free apps, even if they are free, if they offer in-app purchases, your payment method must be in place and valid on the phone to download them.

Are you sure you're signed in?

Are you sure you have space available for install?

That's just a place to start.

What information is it saying about your card?