Can I download without a payment method?

Yes, I do, but only under very restricted conditions. 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 - https://support.apple.com/HT207165

- "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' - https://support.apple.com/HT203005

Create or use your Apple ID without a payment method - https://support.apple.com/HT204034

- "You can remove payment methods from your existing Apple ID after you sign in to the App Store, iTunes Store, or Apple Books. You won't be asked for a payment method again until you make a purchase." However, "To make purchases from App Store, iTunes, Music, and more, you must add a valid payment method to your Apple ID" - https://support.apple.com/HT208650 In this cases purchasing means something that costs money. As illustrated by the instructions on how to create an Apple ID without a payment method on a computer, it actually has you download a free item without a payment method, showing this is possible.

If you can't remove your last payment method or use no payment method with your Apple ID - https://support.apple.com/HT203905

- " if you're using an existing Apple ID with the App Store for the first time, you must provide payment information."

- "If you have any active subscriptions, you must have at least one payment method on file other than Apple ID credit."

- "You can't remove a payment method if you have a balance due."

Also:

How App Store and iTunes Store purchases are billed - https://support.apple.com/HT201359

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.

If you still have problems Click here --> https://support.apple.com/choose-country-region/media-services 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." - https://support.apple.com/HT201232