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 desktop or laptop computer click on and download
Download the com.overdrive.mobile.android.libby.apk file and make note of where you save it on your computer.
Step #3: Transfer downloaded file from computer to Kindle Fire
Unlock your Kindle Fire to access the homepage of your device. Attach the Kindle power cable to power port on Kindle Fire device and plug the USB end into a USB port on your computer. Open File Explorer on your computer and navigate to the folder/location where you saved the .apk file that you downloaded in previous step. Copy the file, and then from This PC in File Explorer navigate to Fire > Internal Storage > Download, and then paste the .apk file.
Step #4: Install the Libby app
Unlock your Kindle Fire and go to Docs > Download > Local Storage and tap on the .apk file. Follow steps to install the app.
Step #5: 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 #6: Go back to Step #1
...and disable the setting "Enable Apps from Unknown Sources."