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 having others make sure other family members have adequate personal balance 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 (but not the Organizer's personal account balance).
From: How to share purchases with your family - How to share purchases with your family - Apple Support
"When you turn on purchase sharing, everyone in your family gets access to apps, music, movies, TV shows, and books that family members buy. The family organizer is billed for family members' purchases."
Ideally the second sentence would also mention that others can still pay for their own items, they just have to use personal account balance to do so (it is mentioned if you go to the tiny footnote near the bottom of the page and follow the link there). Turning on purchase sharing simply activates the feature whereby if a family member does not have adequate personal balance to pay for something then the organizer's payment method (e.g., credit card) will be charged.
Read the document for the link to "learn how purchases are billed if a family member has Apple ID balance" ( How apps, content, and subscriptions from Apple are billed - Apple Support ). This tells you how it works. In essence the payment method for Family Sharing is automatic:
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 it cannot bill the primary payment method, the Organizer will need to make another payment method the primary 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 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 ID - 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
