Dear Author blog.
According to that post, the GetJar marketplace has the Kindle app, so that was my starting place. In the version actually on the Kobo Vox, it can't seem to be found at all, so I tried downloading it to my computer to e-mail to myself on the Vox. Something they say can be done. According to GetJar, the app in question doesn't work with the Vox, but I've seen all those mentions of having installed it successfully.
So, on to the next step in my quest. Installing another marketplace. Easily done. I tried another of the marketplaces on their list: the Soc.io Mall. Opened up my browser and input the url for that, installed it and registered. They wanted a bit more information than I really wanted to give though. I did it anyway. Success.
Searched for the Kindle app on that marketplace and installed it just the same way I would an app from the GetJar marketplace that came with the Kobo Vox, and installed it. No warnings about it not working with my device. Their install method follows a bit of a different process though. Instead of "Download" the button says "Accept Permissions" and then after that, "Install". From there it either brought up the install screen or I had to bring down the Notifications window and open the downloaded file myself. I honestly can't remember, having been a bit distracted at that point in the process. Either way, it was a simple thing to do.
After the Kindle app is installed, you register your account the first time you open it. If you already have an amazon.com account, use that, or else create a new account. Once you're in, the app comes with three free classics, and you can get more books through the menu button. The process seems to be fairly intuitive.
Edited to add:
The latest update to the Kobo Vox has made this unnecessary now. With the access to the Play store, it's now easy to search for any app you want.