Apple Health (AKA Health) is an iOS app from Apple that allows users to track data collected by HealthKit, which includes things like sleep tracking, heart rate, steps, exercises, and more, which are collected via the Apple Watch and the iPhone.
iOS app to export data
Create an iOS app that has HealthKit permissions and serializes all collected data (or some subset of it) to JSON, then sends it up to a web service which stores it. Then, a user can authenticate with this web service using IndieAuth, choose which bits of data that they want to publish to their site via Micropub, and on what frequency.
Some good tools that may be helpful: