James,
I don't see how an illustrator can submit a book idea to a publisher WITHOUT a few final art samples! I don't think a character sketch is enough. It's absolutely necessary to complete at least a few scenes/spreads of art (with the character in different perspectives, poses, and environments to show consistency in the character throughout). I'm a former children's book Art Director, and currently a published author/illustrator that's been submitting directly to publishers without an agent for 9 yrs. I find showing dummies with complete ms, and full-color art samples the most effective way to submit my work.
If you're talking about getting hired as just the illustrator (and not your own book), then the process is different. You just give the editor/art director a link to your website so they can view your portfolio. But for book subs, you would need completed art samples, with or without agent.
I would not recommend working with an individual on a children's book. These collaborations rarely do well because there isn't the expertise needed to create a quality children's book. I would suggest to work only with publishers, who have the necessary resources in editing, design, production/manufacturing and sales/distribution.
Good luck!