Unable to download free apps without a payment method on iPhone

If it is asking for a payment method then you have no option but to enter one such as a credit or debit card. Read the support article Add a payment method to your Apple Account - Apple Support about how to do that.

If you do not have a credit card, a debit card may be acceptable instead. See Payment methods that you can use with your Apple Account - Apple Support Note that that article can be a bit confusing because it may say you can use account balance, but for some items you must have a credit or debit card present even if account balance will be charged first.

If you do not have a card, see if somebody is willing to add you as a member of their Family Sharing group ( Set up Family Sharing - Apple Support ). They will, however, ultimately be responsible for any purchases you make should you not have sufficient personal account balance to cover any charges. If you are not a full adult they may wish to use Approve what kids buy with Ask to Buy - Apple Support to restrict your purchasing ability until you have your own card and can leave the group.

Sometimes a card need only be presented once. If you are using your account for the first time:

Sign in to your Apple Account - 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."

See if somebody is willing to enter their card to satisfy this requirement, then remove it immediately.

If it is insisting that a card be present then you will not be able to make that purchase without one. I have found that items that are completely free do not require a card be present, but I also have a very old account that may have different rules.

A payment method is required because most apps, even the free ones, offer upgrades and in-app purchases. I'm sorry for your circumstances but this is the way it is.