According to their FAQ section:
Is Audiobooks.com mobile friendly?
It sure is! You can stream or download your favorite books from Audiobooks.com on your iPhone or iPad (iOS4.0 or higher), or Android smartphone (running version 2.3 or higher), via our native apps.
Thanks to our HTML5 web app, you can also listen to audio books using almost any other internet-enabled device, like laptops and tablets. We recommend that you check with your mobile provider to make sure you have a data plan.
So if your Kindle supports web browsing (and has an HTML5 compatible browser), then yes you can. Otherwise, you are forced to use one of their native apps for the devices mentioned.