If you're serious about reviewing products, then maybe this might be of interest to read.
First things first: your place of writing doesn’t have to be a professional website. You can be a fan blog and still get review copies — it all depends on how exactly your site is styled, how it looks, and your writing.
From there, getting in touch with someone in your industry of interest for review copies isn’t difficult, it’s a case of whether or not you’ll get a response. It’ll help if you know who want to speak to at a company. Then it comes down to how you present yourself (you have to introduce who you are, your site, and its purpose — your pitch), and you can go from there. If you just started your blog, unless your site gets 5,000-10,000 hits or something sizable, you need to build yourself up for a good three to six months. From there the company will decide what kind of access you get.
The best way to try and get on a press list to meet them at an event, which for anime and manga is a convention. “But they’re probably going to be busy–” Yes, they will. That’s why A) you need to be short and sweet with your request (Hi, my name is so and so, a writer at x site, where we cover seinen manga” is a start, then get into what you want), B) you also have to figure out the best time to approach them, so study up on when their panel will take place, when a con’s going to close, or, C) be press at a convention, which means you’ll be added to press lists for publishers. Not all of them of course, but for some of the big conventions expect to get emails from companies you might not even know about.
But since I started writing in 2011, I introduced myself to people at conventions — it’s convenient to have the name and face & then send that email out, rather than send an email out and hope to get approved. Both options can work, but that’s just my two cents. Otherwise, be responsible with asking for free stuff. And also be consistent too.
You can get a bit more info on how to approach companies from Lauren.
NOTE: If you’ve set up a way for people to get in touch, don’t be surprised if companies reach out to you.
NOTE 2: You should not attempt to send emails every day or once a week if you do not get a response back. At least wait a month before following up. If you get nothing, they may think you need to prove yourself more. So you just need to wait, then try again later — maybe 6 or so months down the line. If still nothing, they may not want to interact with you. It is what it is.