I'm not sure if this is the correct forum but:
The Android-N update for the Nvidia Shield Android TV will include an Amazon Video app. The update should be available in the next week or so, but for those who cant wait I found a link on the Nvidia forums this morning.
https://www.reddit.com/r/AndroidTV/comments/5nw302/amazon_video_extracted_from_gen_2/
I just installed this on my Shield and can confirm it's working.