App installation is handle by iOS and then App Store that is run by Apple. Apple handles the purchasing, provisioning, and installation of apps on your device. If you are having issues with the app installing, make sure you are on a good Internet connection and if the problem persists,
restart your iOS device.
In the interim, you can search for OnSong by swiping down while viewing your device's home screen and search for OnSong. If OnSong does not appear, it may not be be installed. Otherwise you should be able to open OnSong from the iOS search screen.