Is it possible to approve a Family Sharing purchase on a Mac? I have family with iOS devices (iPhone, iPad) but I personally use Android.

I do however have a Mac Mini for development - is there any app or web site from where I can approve App purchases (Ask to buy)? The idea is that I won't need to confiscate an iOS device and log into App Store using my "organiser" account just to approve.
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.