Apple certainly doesn't let you.

There are several apps[0] off the top of my head that pull data from Apple's HealthKit. Apple even has a framework for it: HealthKit. That said, I've never written to that API myself, but it observational evidence indicates that you can pull data from Apple just fine. The app I use to pull data off Apple and ship to Strava says as much.

[0] RunGap and HealthFit