The Libby app is not available in the Amazon app store on Kindle Fire devices, but it can be installed by following the steps below.
Step #1: Prepare Your Fire Tablet
Go to Settings > Security (or Applications) > Enable Apps from Unknown Sources (this will prompt a warning message—that’s normal).
Step #2: Open your device's Silk browser
Visit https://apps.goodereader.com/android-apps/e-reader-apps/?did=39495 where you'll find the Libby app. Tap on the Download button.
Step #3: Install the Libby app
After the download is complete, tap Open > Install to install the Libby app.
Step #4: Set up the Libby app with your library card
Launch the Libby app from the Kindle's carousel page and follow the prompts to find your library and add your library card(s) to the app.
Step #5: Go back to Step #1
...and disable the setting "Enable Apps from Unknown Sources."