It does not matter who actually initiates the purchase. In Family Sharing the purchase will always be charged to the Family Organizer unless the initiator has iTunes credit balance in their account to cover the cost of the purchase, in which case it will use that to pay for it.
Once bought by whoever member decides to buy it, it will be available to share from that member's Library if they have Purchase sharing turned on. It will not appear as free directly on the App Store for other members, only available from the Library of the person that started the purchase for other members to download.
See here: link-> How to share purchases with your family - Apple Support