Is it possible to approve a Family Sharing purchase on a Mac?
Yes. This is possible. The instructions for all the necessary steps can be found here, provided you have OS X 10.10 Yosemite or higher. When someone in your family requests to make a purchase, you will get a notification on your Mac. (You can find past notifications in the Notification Center). You can approve the request directly from the notification, and it will ask you to authenticate with your Apple ID credentials.