For a purchase not to be charged to the organiser's credit card that member needs to have a balance on their account : Family purchases and payments - Apple Support
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. As the family organizer, any receipts generated by the transaction will be sent to you.
So their own credit card won't be charged, it's their balance that will be used (if no balance, or any remainder if their balance doesn't cover the entire price, the organiser's credit card).