This app was mentioned in 5 comments, with an average of 2.00 upvotes
If it's a tablet, you could also use Flixter app. You log into it and link your VUDU account, it will allow you to download your movies to an SD card for later viewing. I'm not sure how it does it, i've never found the obvious files, but it does work. I have used it on my phone before.
Best way would be to install Google Play services so you download regular Android apps in addition to those in the Amazon store. Then you can probably install the Flixter app just fine.
I followed the "How to Install Google Play on Fire Tablets" section in this article a couple weeks ago and it turned out great. You may want to try it out.
https://play.google.com/store/apps/details?id=net.flixster.android&hl=en
Says it's compatible with all my Android TV devices