Flappy Bird was never available in the Amazon Marketplace and will most likely never be available since the app was officially taken out of the Google Play and Apple market. Fortunately we can still sideload the game to our Kindle Fire. The method will work for all versions of the Kindle Fire including the first generation, the HD, and the HDX.
How-to Sideload Flappy Bird to the Kindle Fire
1. Enable Installation of Applications from Unknown Sources
For the Kindle Fire & Kindle Fire HD: Go to Settings > Device > Enable ‘Allow Installations of Applications’
For the Kindle Fire HDX: Go to Settings > Applications > Enable ‘Apps from Unknown Sources’
2. Open the Silk browser and go here to download the Flappy Bird .APK file: http://www.mediafire.com/download/g1kkd5v72k37r2k/flappybird_1.3.apk
3. Open and install the file
4. Enjoy the insanely hard and addicting game!