I just found these instructions for installing the Zinio magazine reader on the Kindle Fire. They worked fine for me.
Temporary alternate download method for the Zinio app on the Kindle Fire
Note: these instructions are ONLY for users on the Amazon Kindle Fire. This installer is not supported for any other device.
- From the main screen, tap Settings > More > Device > Allow installation of Applications (from unknown devices)> select “ON”
- Download the APK (app installer) from here .