Dropbox isn’t available in the Amazon App store. You’ll need to make a couple of adjustments to your Kindle Fire to to install it. You don’t need to root your device to install it.
First tap gear icon at the top of your Fire then select More.
Next, select Device from the Settings menu.
Set Allow Installation of Applications to On.
Now, launch the web browser and type: https://www.dropbox.com/android into the address bar. Then tap Download App.
Give a few seconds to download then tap the notifications next to your Kindle Fire’s name. Then tap the Dropbox.apk.
You’ll see a description of the Dropbox app.
At the bottom the the description screen tap Install.
After Dropbox installs, tap Open at the bottom of the screen.
Tap through the Dropbox tour screens, the log into your Dropbox account.
Now you have access to all of the files stored in your Dropbox.
You can upload, share and manage files in your Dropbox account just like you can from any other device or computer you have Dropbox installed.
The Dropbox icon will show up when you go to the Apps section on your Kindle Fire. If you want quick and easy access to it, pin it to Favorites.