How can I continue without a payment method to download free apps

Apple requiring a payment method such as a credit or debit card, even for free apps, is not an uncommon observation. Some possible reasons are outlined in the links below such as outstanding balance, an issue with a payment method you have currently entered, or ongoing subscriptions. You can try to set up your account without a payment method but you may find there is no option other than to enter a valid payment method to satisfy Apple's requirement. If the item is free you will not be charged. If has a cost, your Apple ID balance will in most cases be used first. You can also later remove the payment method and see if it only needs to be there temporarily.

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

- "You might need to have a payment method on file, even if the app that you want to download is free." If you have something else on your account that requires a payment method, such as a subscription, and your current payment method is not working then Apple won't let you download even free things until you fix the payment method. This is explained in this document, 'If your payment method is declined in the App Store or iTunes Store' - If your payment method is declined in the App Store or iTunes Store - Apple Support

If you can't remove your last payment method or use no payment method with your Apple ID - 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.

Also:

How App Store and iTunes Store purchases are billed - How apps, content, and subscriptions from Apple are billed - Apple Support

Some purchases aren’t charged to your Apple ID balance

- 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.

So the resolution will be to add a payment method:

This document explains how to change, add, review, or remove Apple ID payment methods on a Mac or Apple mobile devices; click here --> Add a payment method to your Apple ID - Apple Support

If you still have problems Click here --> Choose your country or region - Official Apple Support Select your country, then a product. If you don't see one that handles your issue then keep experimenting with selections until you reach one that gets you a chat session or a telephone call and get the representative to redirect you.

or:

Contact Apple for support and service --> "See a list of Apple phone numbers around the world." - Contact Apple for support and service - Apple Support