I am afraid to inform you that the Amazon Music app has a newer, minimum iOS version, now of iOS 12 and later.
The app is no longer compatible with your over 8-year old, 2012 iPad 4th generation.
Here is what has happened.
Virtually, all third party iOS app developers ended/stopped/abandoned supporting these old 8-10 year old iPads over two years, ago.
Third party iOS App developers simply do not want to continue to support 8-10 year iOS devices, any longer, and have permanently removed ALL older version apps from the iOS App Store.
There are virtually, NO MORE older iOS 9 or 10 compatible apps available from the Apple iOS App Store, any longer.
There is NO LONGER ANY WAY to retrieve or initially download older compatible apps from Apple’s iOS App Store.
Simply put, your 4th gen iPad is too old and obsolete now. and has been so for the past two years.
Sorry.
If you need/want an iPad with an up to date iPadOS with access to current, updated apps, then you really need to seriously consider a new iPad model OR a much “newer” , used iPad model capable of running the latest iPadOS versions.
There are plenty of newer, older iPad models that can run the latest, current versions of iPadOS.
Sorry & Best of Luck to You!