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Kindle or Kindle Fire?
My public library gives me access to all the digital copies for free (well, $5 for the library card plus my taxes...) and they do it through the Zinio app which you can get for Kindle fire.
Have you tried this? [link]
Just like clicking "Get the free Kindle app"? Maybe that'll do it for you. I'm not sure about a regular Kindle.
You can subscribe to them via Amazon. For example, this one is here: [link]