Yes, I believe you need to have a payment method on file.
Even a lot of "free" apps these days are like arcade games. You can use some features for free, but progressing past a certain point may require (or be much easier with) the purchase of artificial consumable in-app purchases. You won't be charged for downloading a free app – but you will be charged for those.
In some countries, the payment method can be an Apple Balance, and you can load the Apple Balance using gift cards that you can purchase at many retail stores. If you set things up that way, I don't believe Apple would have your credit card number.
Some stores sell preloaded Visa and MasterCard debit cards, and those might be acceptable payment methods. But the banks/vendors who sell those cards add a lot of fees. At least buying Apple gift cards and using them to load up an Apple Balance wouldn't involve those.
Payment methods that you can use with your Apple ID - Apple Support