Unfortunately, Apple no longer supports older versions of iOS. They don't let you download old versions of the app through the iTunes store, and they also have restrictions that don't allow new apps to be submitted that support both old and new devices fully.
In addition to Apple's restrictions, some of the updates we've added recently and will be adding soon use features that aren't possible on older versions of iOS without significant circumlocution of code.
If you join the "LDS Music for iOS" project (see the "Projects" tab at the top of the page) you can access the following page, which lets you download beta versions of the app – including older beta versions from back when iOS 4 was supported:
http://tech.lds.org/dl/ios/
iOS 4 devices should be able to run LDSMusic-Staging 1.3.1 (1193) or anything before that.