The Amazon App store doesn't provide the official Instagram client. This slide will show you how to isntall Instagram on Kindle Fire.

1. Get Instagram on Kindle Fire
www.epubor.com

2. Many social network service have launched dedicated client for
cellphone and tablet. We can easily enjoy Facebook and Twitter on
our Android phone and iPad. But for Kindle Fire users, they may
not enjoy this convenience as the Amazon App Store is always lack
of apps, even the popular ones. Instagram is the example. There is
no official Instagram client for Kindle Fire but only a simplified
version called “Instafire” which even don’t let you post pictures!
But actually it’s absolutely possible to install Instagram on Kindle
Fire. This article will show you how to do this job.

3. First, you need to allow your Kindle Fire to install third party apps.
Go to “Setting” – “Device” and enable “Allow Installation of
Applications”

4. Open the Silk browser and visit
download.pandaapp.com and
type “Instagram” in the search
box. Click “Search” and you may
find several results. Select the
right one. Pay attention the app
name and version

5. Click to go to the detail page.
Tap “Download” – “Fast
Download” then the browser
will start to download this app.
Tap the top of the screen to
check the downloading
progress.

6. You will meet a request of installation after downloading finishes.
Confirm to install the app and then you can use the Instagram on
Kindle Fire.
Source: http://www.epubor.com/get-instagram-on-kindle-fire.html