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: Using your device's Silk browser
Step #3: Install the Libby app
Choose Open in the lower task bar to start the installation process and then choose Install.
Step #4: Set up the Libby app with your library card
Once the installation is complete, you can choose Open again or simply visit your device's Home screen to launch the Libby app from there. Follow the setup 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."