How to install Android apps without having Google Play?
What you're referring to here is called sideloading - this is a term used when loading an app on to a device without using the/a market store.

It is possible to get APK files for apps, but there're generally good reasons why most developers don't make these publicly available - some developers will make these available thought, it depends on the individual.  

I would email the developer in question and ask if you could have an APK version of the app.  I wouldn't imagine that they would have any issues with this if your request is legitimate and above board.