One of the features coming in iOS 11.3 will let you use Face ID for family sharing. This includes approving purchases. It’s already shown up in the iOS 11.3 beta (via 9To5Mac).
Face ID for Family Sharing
You have to enter your password the first time you get a purchase approval request. But after that, iOS asks if you want to enable Face ID for future purchase approvals.
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Then, when a new request is made, tapping the Buy button will launch Face ID authentication instead of a password prompt.
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