Edited in response to a question in the comments as of January 2, 2012
Here's the up-side though, Overdrive, the makers of the app have it linked from their website, bypassing the Android Marketplace completely. Thus, it's a breeze to download to the Kobo Vox and install.
I googled Overdrive, then found on their site the Overdrive Media Console, selected the operating system (Android), chose to download it directly from them, and then agreed to the user agreement. From there, it took less than a minute to download.
To install, I've found that the best way is to bring down the notifications screen and then touch (I keep wanting to say "click") the file you've just downloaded. From there, follow the prompts.
The newly installed Overdrive Media Console app can be found under "All Apps".
Once you've opened Overdrive Media Console for the first time, the shelf will be empty except for the welcome option. You'll have to choose your libraries, which is one of the options you'll get in the menu from the button at the bottom of the screen. Go to "Get Books", and it will ask you to choose your libraries.
You can either use your postal code or browse for libraries. I tried the former and got told there were no results which I know is not true, as I've borrowed for my old Kobo before. So, I'm browsing to find libraries. It will bring it up by country. Narrow it down, until you find your libraries in the list. Save the library, and start browsing or searching.
Once you've found a book you want, add it to your cart and either keep searching for more books or proceed to check-out. Apparently, you can have a maximum of five titles out at any one time.
The first step in the check-out is to choose the library that issued your library card from their drop-down menu. Then, you have to type in your library card number. I found that doing it with no spaces worked. Choose your checkout period and confirm the check-out. Hit download, and now discover a quirk you didn't know about. You need to have an Adobe ID to use the book. Easy enough to get, and you might already have one.
The books you borrow will be stored and accessed within the Overdrive Media Console.
If you've checked out books on your computer, you can still get them onto the Kobo Vox:
First go to the "Get Books" option in the menu. Select the library you checked the book out from and go to the "My Digital Account" option.
Go to the My Bookshelf option and from there you can download the books you've checked out.