The way it works currently is any app developed in house can be deployed OTA, but third party apps must be installed through the App Store. Enterprises that setup a MDM solution to manage their iOS devices usually choose something that comes with a feature to install a WebClip on the devices. This WebClip will open a website (maintained by the tool) and will display a list of "recommended" applications for users, any of which they can tap and will be brought to the App Store page for that app. All the user has to do is tap the Install or Buy button using either their own iTunes account or a shared company iTunes (depending how large the company is, this may not be feasible). Apple has recently rolled out a volume purchasing program for education institutions only. This means they can buy in bulk and deliver these to their devices, I believe OTA, but not positive on that one. Apple has within the last few weeks announced they will be extending the program to businesses as well, but not a timeline in which they will do it. This may allow bulk purchasing and delivering OTA, but again not positive on that point.
With iOS devices, one of the very common strategies you will see in the enterprise, is what they call the carrot and the stick approach (I did not coin the term). The list of required and recommended apps is published to users, and the installed applications are monitored. If a required application is not installed, then they have the MDM solution deny the device access to corporate resources until they become "compliant." This strategy is used for other things the company wants to "require" as well.