My company has an app that can be downloaded through a website using Enterprise Distribution.
In iOS 9, tapping the download link brings up the expected pop-up "(website) would like to install (App Name)". Tapping 'install' on that dialog first causes the app to get stuck "Waiting..."
App stuck waiting
Until it finally indicates: "Unable to Download App (app name) could not be downloaded at this time."
Unable to Download the App
Installing works as expected using iOS 7 and iOS 8.
How do we support Enterprise Distribution in iOS 9?
Apple has silently changed how Enterprise Distribution works several times, so knowing the answer to this question will not actually benefit you in the long run. How to find the answer to your question the next time will benefit you most.
Connect your device with the shiny new iOS version to your Mac running Xcode, and open the Devices organizer. Select the device, and optionally clear the log. Now open your app distribution page and try to install the app (clear any previous versions of the app off the device first), and keep an eye on the log output. Here you will find hints as to what you need to change.
For iOS 9.1, I saw the error: BundleValidator: Failed bundleIdentifier: com.example.appName.XXXX does not match expected bundleIdentifier: com.example.appName, which means that the bundle-identifier in the project plist file did not match the Bundle Identifier from the project. I remember adding .XXXX to the bundle identifier in the plist file last time I had to fix Enterprise Distribution, for iOS 8.0, i think. Fixing this was easy, just remove the offending .XXXX -- but then it dosen't work for iOS < 9.0
For the time being, I maintain two plist files for distributing the same ipa file, one for iOS 8.+, and another for iOS 9.+