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.
Or another option as mentioned by DVK in the comments, you can turn your Kindle into a generic Android device. It's possible to do this because Kindle devices (and Nook for that matter) are using Android under the hood. Amazon (or Barnes and Knoble for the Nook) just design the stuff on top of Google's open-source operating system. Once you convert your device to a generic Android device, you can run the native Android app provided by audiobooks.com. As for how to change your device, if you choose that route, there are many tutorials out there. Here is one to get you started.