I believe if you sideload the play books app you can then do this, however I haven't tried this myself yet.
The first thing I'd try (assuming you have another generic android device). I do not know if this works and I am not responsible if this causes problems for you. Please proceed with caution. I have personally not had a problem from testing things like this that I couldn't solve without simply uninstalling the things that I tried installing... but your mileage may vary!
1. Download the play books app and the play services on your other android device
2. Download the EFS file explorer app on the other android device
3. Backup the play books app on the other device, you may also need the Google Play Services piece
4. Copy the files over to you computer, and then plug you fire into the computer and copy the files on the fire
5. In configuration of the fire change the switch to allow loading of apps from other sources (if you haven't already)
6. install both files and see if it now works. I use EFS File explorer on my fire to run the files I copied onto the fire. I'm sure there are other ways...