Yes, it's built-in (there's a "Videos" link at the top, and any videos you start watching show up in the carousel interface).

I've seen an APK for Amazon Instant Videos floating around the XDA forums. It's too buggy to be truly usable on anything but a Kindle Fire but it's definitely a stand-alone app. If you've rooted your Fire, you'll see it sits in your app drawer like any other app.