Yes! The cloudLibrary app works on Kindle Fires. Here is a link that will walk you through installing the app on your Kindle fire:
http://www.yourcloudlibrary.com/index.php/en-us/get-the-app/kindle-fire, opens a new window
And if you prefer video, here's a short clip that walks you through the process:
If you have an eInk Kindle you will need to use the Overdrive Woodlands Downloadable Library to access eBooks.