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.
=Full details=
Based on: How to share apps and purchases with your family - Apple Support and How apps, content, and subscriptions from Apple are billed - Apple Support
1. If anybody in the family buys or subscribes to something, Apple first attempts to charge the item to that individual's Apple Account (Apple ID) balance, if any.
2. If a family member does not have enough personal Apple Account balance, any excess will be charged to the Family Organizer's primary payment method (usually a card of some kind). The Organizer's personal balance will not be used for purchases made by other family members. If Apple cannot bill the primary payment method, the Organizer will need to make another payment method the primary method. With Family Sharing the Organizer always has to have at least one functioning payment method.
Also note: "Some purchases, including gifts, can't be billed to Apple ID balance and will be charged to the family organizer's payment method." "Some subscriptions might not be charged to Apple ID balance."
Ref:
- Family purchases and payments - How to share apps and purchases with your family - Apple Support
- How apps, content, and subscriptions from Apple are billed - How apps, content, and subscriptions from Apple are billed - Apple Support
- Check your Apple ID balance - Check your Apple Account balance - Apple Support
- Add a payment method to your Apple ID - Add a payment method to your Apple Account - Apple Support
Adding funds to your Apple ID balance:
- How to redeem your Apple Gift Card or App Store & iTunes gift card - Apple Support
- Add money to your Apple Account balance - Apple Support
