Can AppleOne family members set up their own payment methods

Family members can use Apple Account balance for their purchases.

And as long as the family member has money in his/her own Apple Account balance, for example by redeeming Apple Gift Cards, Apple will subtract purchases for this family member from his/her Apple Account balance, before the organisers payment methods are being used.

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When the family organizer turns on Purchase Sharing and the members enable the feature for their devices, they get access to each other's shared content. The family organizer pays for everyone's purchases unless they turn off Purchase Sharing or other adult family members use their own payment method to make purchases.*

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* If you're in a Family Sharing group, your purchases are charged to your Apple Account balance. If you don't have enough balance for the purchase, the remainder is charged to the family organizer if Purchase Sharing is turned on, or you can add your own payment method.

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The family organizer can also: 

• Update their payment method or add a new one.
• Check purchase history to see purchases made by family members.
• Require that children in the family group request approval for purchases and free downloads.

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copied from: How to share apps and purchases with Family Sharing on your iPhone or iPad - Apple Support

Note that the reply by Ingo2711 describes the long-standing payment system, but there have been changes this year. The basic answer is to your question is yes, adult members can now have their own payment methods under Family Sharing. Read this support article: About iOS 26 Updates - Apple Support - describes the iOS 26.7 addition whereby an adult individual can add a payment method on an iOS device. "Purchase Sharing lets adult members in Family Sharing groups use their own payment method when making purchases, without relying on the family organizer."

I don't use the latest macOS version so I don't know if this is still an iOS-only feature.