Instructions to download the Zinio magazine app for Amazon Kindle Fire is mentioned in the right column of the Zinio help page

Zinio has an app for Amazon’s KINDLE FIRE! –> that’s linked to with specific instructions:

Temporary alternate download method for the Zinio app on the Kindle Fire

Some users are reporting an inability to locate the Zinio app download page from their Fire. Users are encouraged to contact Kindle support here to find out why it isn’t available for download, despite being approved for the device. 

Note: these instructions are ONLY for users on the Amazon Kindle Fire. This installer is not supported for any other device. 

1. From the main screen, tap Settings > More > Device > Allow installation of Applications (from unknown devices)> select “ON”

3. Select “Notifications” to monitor your download. Once the download is complete, select “Install” and press “Open” when the installation is complete.

We hope you enjoy the app! We apologize for any inconvenience you may have experienced when trying to install the app.

Zinio Customer Support

aside: to download the Zinio app for other Android devices, it’s in the Amazon App Store