Icinga Director provides its own REST API where you can manage objects and start deployments.
You’ll find the documentation here; feel free to share your own findings and experiences.
For full automation tasks, Icinga Director can be integrated into existing tools as described in the documentation. Meaning to say, you can easily deploy Icinga Director with Puppet & others.
For more insights, we encourage you to watch existing videos from OSMC and other events.
You may also have a look at the Webinar Director Videos (German language only).