I only have completely free apps. I know I personally can download and update these apps without having a payment method, or even any account balance. However...
Apple's official support articles present multiple statements about requiring the presence of a payment method at times, even for free items. If you are encountering one of those situations then based on the documents I suspect you have no option but to enter a valid payment method if you wish to continue downloading. Read the support article Add a payment method to your Apple ID - Apple Support about how to do that. The payment method may not be charged if you also have account balance or the item is free, but the payment method has to be present. This next article lists valid payment methods for different countries but note that in this situation Apple Account balance alone apparently may not meet the requirement and you need to use another payment method until you find one that satisfies Apple's requirement (often a credit or debit card): Payment methods that you can use with your Apple ID - Apple Support .
If you have already entered a payment method, use the second link above to check it. If a message appears in red there is a problem with it. If the information is still correct and current, contact your financial institution. Also check this resource --> If your payment method is declined in the App Store or iTunes Store - Apple Support
* Ref.:
Add a payment method to your Apple ID - Apple Support
"To make purchases from the App Store and use subscriptions like iCloud+ and Apple Music, you need at least one payment method on file."
If you can’t download or update apps - Apple Support
- "You might need to have a payment method on file, even if the app that you want to download is free."
Remove a payment method from your Apple ID - Apple Support
- If you have any active subscriptions, you must have at least one payment method on file other than Apple ID credit.
- You might not be able to remove a payment method if you have a balance due.
Sign in with your Apple ID - Apple Support
- "When you sign in to the App Store and other media services for the first time, you might need to add a credit card or other payment information."
How apps, content, and subscriptions from Apple are billed - Apple Support
- You can’t charge gifts to your Apple ID balance. If you send a gift or gift card, your primary payment method is charged.
- Some subscriptions might not be charged to your Apple ID balance.
What you can buy with your Apple Gift Card or Apple Account balance - Apple Support
"Some purchases might require that you have a credit card on file, even if you use your Apple Account balance to make the purchase."
