According to this PC Advisor story, Amazon is planning to launch an Android app for its video streaming services "soon." The article is linked below, but the gist is Russell Morris Amazon Instant Video's marketing director told PC Advisor at their Christmas Party - which according to commenter/Mod Matt Egan is a party to go over products that will come out during Christmas season, which is why it was being held in July.
In the meantime, I just today learned of FlashFox, a Firefox like browser (but not from Mozilla) that has a built in flash player, that works for streaming Amazon Instant Video from their website (you will also need to install the add-on Phony, set it to Firefox Desktop, and check the "request desktop" box). It worked on my Nexus 5, but I haven't tried my Nexus 7 yet since I'm at work and it's at home. There's a free and pro version.
Original article on PC Advisor:
And here's the Phandroid article I first read about it on: