Can family members in Family Sharing have their own payment method?

They can have a payment method added to their account, but it won't be used if Purchase Sharing is turned on: link-> Set up Family Sharing - Apple Support

If you turn on purchase sharing, you'll be asked to confirm that you agree to pay for purchases initiated by the family members you invite and that you have a valid payment method on file.

With Purchase Sharing tuned on, Family members can purchase their own items by redeeming iTunes gift cards onto their accounts. with that done, any purchases made by them will first be charged to their gift card balance on their account before charging the Family Organizer's.

If purchase sharing is turned off, it will use their own Payment methods beyond the gift card balance.