I am unable to purchase any apps from the app store on iPhone 14. I get a prompt to double click and then just get the prompt again.

Hi Tom Bilbro

Welcome to the Apple (user-to-user) Community.

Re: Any thoughts on what is wrong?

Wondering if you have any restrictions on purchases set:

Eg: Apple support have an article online that may help check:

Manage App Store purchases, subscriptions, settings, and restrictions on iPhone – Apple Support

or

Seems, for some app purchases etc., Apple requires we have a payment method listed in our Apple ID account, even if we just "get" free updates / downloads.

To see what payment methods Apple can accept in your country or region:

Payment methods that you can use with your Apple ID – Apple Support

How to: Add a payment method to your Apple ID – Apple Support

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More troubleshooting if needed:

If you can’t download or update apps on your iPhone or iPad - Apple Support

Re: "Face ID works fine for everything else"

You might want to check if there's another setting in place:

Require a password for App Store and iTunes purchases - Apple Support

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If needed:

App Store Support has a website online.

There is a "Get Support" link at the bottom (website link below)

App Store - Official Apple Support

All the best :-)