Since all my current credits are in media tie-in writing, I'll also take a stab at this.
It's not an impossible task, but darn tricky. What you'd have to do is find out what publisher/packager (if any) has licensed the publishing rights from the company that produces the anime. Even places like Tokyo Pop have to license the rights from the original Japanese production company before they can translate the manga for sale here in America.
If a publisher/packager has the license, you can attempt to approach them to find out if they would be interested in receiving a pitch. Generally, they have an idea who they want to write for them, but sometimes things change, authors back out, or if you're really lucky, they're doing an anthology and need a number of authors, new and established, to write for it.
Now, even if you do get your foot in the door, writing a tie-in is not like writing your own novel. At all points in the process, the licensing department has final say on anything you do. When I wrote the Gauntlet Dark Legacy novels, I had to submit a story idea, the editor had to approve it, and then Midway Entertainment's licensing department got to hack off on it. Then I wrote a treatment (seven pages describing the book almost chapter by chapter). Again, the editor and licensing had to approve it. Then finally I wrote the novel and then the edits and galleys . . . at any point in this process, licensing could have stepped in, said "Nah, this isn't what we're looking for" and either I would have had to rewrite or else they'd simply have paid my "kill fee" and found someone else to write the book.
See, that's the joys of playing in someone else's sandbox. It's not your property, they're letting you play with their toys. OH, and at the end of the book, you can't have made life-altering changes. You can't kill off Aunt May in a Spiderman story, you can't have Superman marry Lana Lang nor can you kill off Spock. What you are doing in a licensed story is attempting to develop the character further than the movie/tv show/game/comic did or fill in the gaps between episodes to explain why things happpened the way they did.
Only in a very few series, like the Star Wars books, are the books allowed to "continue" the story and do things like kill off major characters (like Chewbacca), but believe me, that was cleared by George Lucas or it would never have reached print.
So, it's not an impossible task, but it's not an easy one either. Best of luck if you want to pursue it and if you have any other questions, I'll do my best to answer them.