Amazon has launched their Cloud Player recently, offering 5GB of free cloud-based storage to users, with additional space available via subscription and 20GB available for free by an album download. Amazon allows for music on their Cloud drives to be played back by users either via a browser, or by way of an app. A platform notably missing from Cloud Player support has been iOS - the app launched on Android as an upgrade to the official Amazon MP3 app, but an iOS app has not been released. Why the Cloud Player has not had support for iOS yet is a good question. The most likely reason is because the Cloud Player is inextricably linked with Amazon's competing Amazon MP3 Store, especially as purchases from the store are available on users' Cloud Player accounts immediately. This likely means that there will never be an official iOS app version for the Cloud Player.
However, hope is not lost for iOS users looking to stream their music from Amazon's cloud solution. Amazon has updated the web version of Cloud Player to support Mobile Safari. This support appears to be unofficial - when trying to launch the Cloud Player from an iOS device, a warning prompts that it is not supported on the user's browser. However, the service loads properly, and allows for browsing of music, and it will play back without issue through Safari on iOS devices. As well, the Cloud Player supports iOS' multitasking controls directly, so it is possible to play and pause while using other apps. Track skipping however appears to not be working, after testing on both an iPod touch 4G and iPad 1G, so there are still some restrictions with the player and multitasking, although tracks do auto-advance while in the browser.
While the native app experience for the Amazon Cloud Player is superior, especially on Android, this does at least present something of a solution for iOS users looking to use the service, if/until Amazon is able to get an app on the App Store for users to use. It is unlikely that it ever will show up, but considering that apps like Rdio exist, it could happen someday, and this might just be a start in that direction.