If you visit Amazon's Cloud Player through the Safari web browser on an iOS device, you'll see that it does in fact now work. You'll first hit a warning page telling you that your browser is not supported, but just ignore that. Click into the music in your drive and it will begin playing. It works flawlessly even to the point where if you get a Push Notification or incoming call, the music will be paused.
When Amazon first launched the player they specifically noted that it was only supported by PCs and Android devices. While Amazon has managed to be the first large player to launch a cloud based player, Apple and Google are planning to launch similar services shortly.