I came to Japan a little over ten years ago with a non-CS degree, an English teaching job, and a desire to do web development here. It took me two years of learning the language and building my portfolio of pro-bono/super cheap sites, but after that I started scoring freelance gigs. I would do anything: servers, design, programming, translation, whatever would pay to get me out of teaching. As time went by I could focus more on what I'm good at, server admin and programming. I've worked for foreign firms, Japanese firms, Japanese firms founded by foreigners in Japan and I've set up two of my own companies here. It absolutely can be done.
That said, you will run into bureaucratic nightmares. I could go on for pages about firms that won't allow developers FTP access, ones that won't allow you to set up WordPress (they're not an official vendor), firms that want people to sign up for their email newsletter, but by phone because they think the Internet is scary and they don't want personal information put into their website, and other stuff that to this day I still sort of don't believe.
But you'll learn how to be patient, get through those situations, and build either a client base or find a job somewhere that makes sense. Because there are great clients and good companies to work for here. It takes time, but if you learn the language and have solutions to problems people have and can show some examples of those solutions in a portfolio, you'll start finding work. Webdevs are a dime a dozen, but good webdevs who can seamlessly work in English and Japanese will always have work here.