When one wonders whether a term follows AP style, it's a good time to dive into the stylebook. Sure enough, this reference to homeschooling led him to discover that the world's oldest and largest newsgathering organization, for better or worse, goes with home school/home schooling.
AP also calls for home-school as a verb, home-schooler as a student and home-schooled as an adjective.
Yahoo! posts its style guide online for free, by the way. But there's no listing for homeschool/home school.