How can I download apps without a payment method?

Unless you set up your Apple ID originally to not use a payment method, there is no way around entering a valid payment method now. If you want to download any content at all, you must enter a valid payment method. After you do that, you can use Apple Gift cards or just download free content so that your card is never charged.

Some worthwhile reading.

Create or use an Apple ID without a payment method - Apple Support

Change, add, or remove Apple ID payment methods - Apple Support

You need a payment method to download paid or free apps.

Apple takes your privacy and information seriously, you can be assured that you will not be charged anything without your consent.