How do I get a job as a headphone reviewer?

You can just contact the publisher/editor via email for sites like Soundstage, Part Time Audiophile, Stereonews.net, etc. and tell them you’re interested in reviewing headphones and a bit about your audio experience/equipment.  They’ll probably ask for a writing sample, or in my case they just had me write a review of one of my components and that’s how I got started.  Hope this helps, and best of luck.

Do it for free and attract a following is how social media works today.  

With academia, I would write something related to my discipline and then follow the submission guidelines for whatever periodical I wanted to be published in. So I would think you could do a review and then contact Stereophile or whoever and tell them you’re interested in sending them your work for publication and ask them for instructions on how this is done. Every periodical might have a different set of guidelines. Some editors may welcome these types of entries and some may not. I would not write a review and then just send it in to the magazine to see how they respond because administrative staff will object to this and it could ruin your chances to publish with them altogether.