Currently I believe you can only Stream your content using their Flash Player (small text below the Install Silverlight message). There is hope for a Player as Amazon has an HTML5 Player for the Xbox One.
If you’re an eligible Prime member, look for the Included with Prime or Prime categories on the Amazon Video home page to find movies and TV shows you can watch at no additional cost.
You can stream the first episode of selected TV seasons for free on compatible Amazon Video devices:
Fire tablets (except Kindle Fire 1st Generation)
PlayStation 3 & PlayStation 4
iPhone, iPad & iPod touch
Note : First Episode Free titles include advertising breaks before and during video playback.