The short answer is only one person, the Organizer, can (and must) have a card, but the Organizer can avoid having this charged by others by asking them to make sure they have adequate personal balance (e.g., from gift cards) to cover their purchases. Other family members can only pay for things by using their personal balance or by letting the remainder automatically roll over to the Organizer's payment method (such as a credit card but not the Organizer's personal account balance).
The alternative is described in this document: Add a family member to your shared subscriptions - Apple Support
"If you're the family organizer and you don't want to pay for family members' purchases and subscriptions, you can turn off Purchase Sharing [ How to share apps and purchases with your family - Apple Support ]. When Purchase Sharing is turned off, everyone must use their own payment method. When Purchase Sharing is turned off, family members can share subscriptions, but they can't share individual purchases like apps or books."
Another alternative would be for a person to leave the family group altogether. However, a person can only join one family per year.
How to share apps and purchases with your family - Apple Support and How apps, content, and subscriptions from Apple are billed - Apple Support