Yes, you can even though Amazon didn’t reveal that information during the announcement of the device. In a mashable.com article this has been revealed lately. According to the article:

Although Ama­zon goes out of its way to hide all traces of vanil­la Android from its new Kin­dle Fire tablet, it turns out tech-savvy users can still install select third-party Android appli­ca­tions out­side the aus­pices of the Ama­zon App­store. These apps include Ama­zon com­peti­tors, like Barnes & Noble’s Nook app for Android.

Here are the steps to follow to install third-party apps: