Can you use Non-Apple Gift Cards to buy things on the App Store?

It is not possible to use other gift card, and you need to buy Apple Gift Card or App Store & iTunes gift card.

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

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

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) 

When redeeming a gift card, the amount will be credited to your Apple Account balance and you need to set your Apple ID balance as your default payment.

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 have a valid payment method on file, instead of buying or order an App Store & iTunes gift card, you can Add money to your Apple Account balance anytime, no gift card needed, see -> Apple Gift Card - Apple (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

Yes, If you have enough money in your Apple Account balance, all you have to do is -> you need to set your Apple Account balance as your default payment.

See -> Add a payment method to your Apple ID

Add a payment method to your Apple ID – Apple Support (UK)

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)

and ⤵️

• 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 have followed all the steps and instructions and 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 or check -> here for more support and assistance.