although you can search for digital download/VideoOnDemand content through the Amazon application, it will not allow you to purchase one. "this mobile application does not currently support digital downloads."
however, i have gone to Amazon's full website (via Evo browser) and logged into my account--and watched a Video-On-Demand which i had purchased previously. it worked very well, no audio/video lag and have since considered buying many more movies from the VOD service so i could access them while waiting at an airport or something with a strong wifi signal nearby.