I have also seen some businesses refuse to take a prepaid cash card, some saying nothing,
and others saying that they can not process such a card as recurring billing is blocked by the issuer or something.
I think there is some distinction at the card kiosks re-loadable vs non-relaodable making a difference with some of these issues.
Of course using a loyalty card during purchase or using a debit card to buy one adds personal info into multiple databases I would assume.
In that case, you may have to provide an address on a website.