Home school is on its way to becoming a compound, but it is still only rarely spelled as a single, unhyphenated word. Give it a few more decades. For now, when home school functions as a noun, it is still two words, without a hyphen. When it functions as a verb, it usually takes a hyphen—e.g., I home-school my children; my children are home-schooled.
Of course, words that are compound can’t become so without trailblazers. There is almost no doubt that homeschool will someday gain broader acceptance, so use it if you don’t mind confronting your readers with the new form. You’ll be doing a favor for future English speakers.