Hi JW Stew.
Yes, it can be done. Each person needs to buy iTunes gift cards, and redeem them to their own account, so that they are always keeping a credit balance.
This will cause each person's purchases to be charged against their own store credit balance rather than against the Organizer's credit card.
For more information, see this document: Family purchases and payments
"After you set up your family, any time a family member initiates a new purchase it will be billed directly to your account unless that family member has gift or store credit. First, their store credit will be used to pay the partial or total bill. The remainder will bill to the family organizer's card."