Is there an API or anything that would enable me to write a Windows 10 app to log into my account and receive and raise those notifications on my Windows Phone and allow me to accept or decline them?
No. There’s no developer access to the Family Sharing infrastructure, neither natively on iOS, nor via any sort of web service.
Share and Enjoy
Quinn “The Eskimo!”
Apple Developer Relations, Developer Technical Support, Core OS/Hardware
let myEmail = "eskimo" + "1" + "@apple.com"
That's a bit annoying but understandable even email based notifications/acceptance via the icloud address would've been fine, but it seems to just purely be iOS and MacO/S based devices.
So we're a bit stuck without a "parental" apple device...