Microsoft announced today from the Outlook Twitter account that the app is now available on Amazon's Fire OS. This includes Fire tablets as well as the Fire Phone, but Fire TVs are not supported.
Despite the company announcing it today via Twitter, the app has been in Amazon's Appstore since October 18, and it hasn't been updated since then. The version number is just a bit behind its sister app in the Google Play Store, coming in at 2.1.92, while the other comes in at 2.1.98.
Microsoft also announced recently that there is a new scheduling assistant coming to the Outlook mobile applications. The feature is already available for the iOS app, but it's been promised to Android and Windows 10 Mobile, so you'll likely see it on the Fire OS app eventually.
Amazon's Fire OS is a forked version of Android, so if you're looking to see what updates might be coming to this app, you really only need to look across at what's planned for the one in the Google Play Store.