Ooh, interesting idea...
Well, there's no reason it isn't possible - question is, is it practical? I think the trickiest aspect here would be the length issue since a song's generally much shorter than a piece of prose. BUT that doesn't mean it can't be done... if you plan on writing a fairly short story, yeah, there's no reason you can't write it in song format (after all, aren't songs a form of storytelling?) Or maybe you could combine prose and songwriting, such as writing out the story first and then creating a song about it (or vice versa), or you could start every post/scene/chapter with a line or two for your song, and when you're done, you'll have the finished product. So, yeah, I'd say it could certainly be a practical format for your story if you do a bit of planning before writing the first draft or first post.
If you do wind up doing this, I'd definitely be curious to see the finished product, by the way; this seems like a really cool concept.