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.

I was thinking about your question and realized that headphone reviews are all about the equipment you to drive them. So, I went to see your virtual system. Surprisingly, there isn’t one. I would start there. Let’s see it and maybe a summary of how you developed it and your history. Sounds like you have heard quite a few.. 
 

I’ve used headphones extensively since the ‘80s, and particularly over the last twenty years while putting together a good headphone system. I found the sound of headphones vary greatly depending on the equipment you use to feed them. I have gone through over half a dozen carefully chosen good quality headphone amps and found that the strengths and weaknesses of different headphones can vary tremendously depending on the headamp… the amount of detail and more subtle tonal balance is strongly varies be the source / DAC. Have you found the same?
 

I guess if I were to try doing reviews I would want at least one high end system and a couple that are more common ones. I noticed that differences between headphones ( I have a bunch, I’m sure you do as well) that seem big with less than ideal input nearly disappear when fed a really high current natural sounding input. 
 

Sounds like you are ideally suited skill wise I’m looking forward to hearing more about your interest in headphone systems.