Well, after more time and effort than I'd care to admit, I've finally found a stop-gap solution to read Kindle books on my PlayBook until the Kindle app appears in BlackBerry's App World. Here's how.
First, if you haven't already, install the Kindle app for your PC or Mac (my instructions are specific to the Mac, but the process should be similar on the PC) and use Amazon's Manage Your Kindle page to send the Kindle book you want to read to your computer.
Now, download calibre ebook management software for your OS and open it. Select "Add books" -> "Add books from a single directory" from the top menu:
Navigate to the directory where your computer stores your Kindle purchases and select the title you want to read. The Mac app stores them in [user]/Library/Application Support/Kindle/My Kindle Content. Note that some files have obvious filenames, while others ... not so much:
Select the newly added book from the list (it should have an intuitive title now) and choose "Convert books" -> "Convert individually" from the top menu:
Select EPUB as the output format and click OK:
After a few seconds, you'll see an EPUB format of the book next to the MOBI version. Select "Click to open" to open the directory where calibre stores its files:
Now that you have stripped your book of Amazon's DRM and converted to a format other ebook apps can read, you need to get it onto your PlayBook. Since BlackBerry still hasn't provided desktop sync for the Mac, I'm afraid you'll need to email the file to yourself.
My research has turned up only two real contenders for EPUB readers for the PlayBook: PlayEpub and EPUB Reader. I wish I could recommend one or the other, but each will do the job and each is kind of clunky, especially if you're used to the Kindle app on another device. Still, each one is only $1.99, so I'd recommend giving both a try and figuring out which one you like best.
To get your book on the PlayBook, open your email account and download the attached file from your message. Now, when you open PlayEpub or EPUB reader and choose "Available books:" -> Downloads or Import EPUB, respectively, each app will find and recognize the file in your Downloads directory. Add the file to your library, open it, and you're reading a Kindle file on your PlayBook!