App Store won't let me download any apps without providing a payment method

You can download free apps, but you may need to have a valid payment method on file, even if the app you want to download is free. 

You can have a credit card / debit card on file and an Apple Account balance.

If you have set your Apple Account balance as your default payment method, your default payment method will be used first by Apple when you need to make a payment.

Free apps has also In-app pruchase.

See -> Buy additional app features with in-app purchases and subscriptions

Buy additional app features with in-app purchases and subscriptions

See also -> Apple Media Services Terms and Conditions

Legal - Apple Media Services - Apple

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See -> Add a payment method to your Apple ID

Add a payment method to your Apple ID - Apple Support

• If you want to change your card, you need to add first the new card, then remove the old card.
• or if you want to use your Apple Account balance, you need to set your Apple Account balance as your default payment.

See also -> Payment methods you can use with your Apple ID

Payment methods you can use with your Apple ID – Apple Support (UK)

fyi: Not all credit & debit cards seem to work with App Store / iTunes Store, try a different one, or preferably a credit card. If your card is not accepted as a paymenf method, you need to contact your bank.

If you don't have a credit card or don't want to use a credit card, you can also use your Apple Account balance as payment method.

See -> What you can buy with your Apple Gift Card or Apple Account balance

What you can buy with your Apple Gift Card or Apple Account balance – Apple Support (UK)

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• You can't use your Apple Account balance to send gifts or gift cards.
• If you use Family Sharing, your family members can't spend your Apple Account balance. Funds that you add to your Apple Account balance are only available to you. Find out more about how purchases are billed to your Apple Account balance.
• Some purchases might require you to have a credit card on file, even if you use your Apple Account balance to make the purchase.

If you still need help, get support from Apple directly -> Get Support or call Apple in your region see -> Contact Apple for support and service 

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Where to buy a physical Apple Gift Card or App Store & iTunes gift card: at an Apple Store or an Apple Authorized Reseller in your region.

See -> How to redeem your Apple Gift Card or App Store & iTunes gift card

How to redeem your Apple Gift Card or App Store & iTunes Gift Card – Apple Support (UK) 

See also:  If you can't redeem your Apple Gift Card or App Store & iTunes Gift Card

If you can't redeem your Apple Gift Card or App Store & iTunes Gift Card - Apple Support

Beware of -> About Gift Card Scams -> About Gift Card Scams - Official Apple Support