Basically — no, they don't. The jist of it:
"No government agency regulates the labeling of mattresses as “organic” or “natural,” and trade groups like the International Sleep Products Association and the Specialty Sleep Association offer their members no guidelines for using the terms."
So, buyer beware, it's marketing, marketing, marketing! Whether your mattress is made from "soybeans, stuffed with coconut husks, infused with green tea and treated with aloe vera" doesn't really matter. The New York Times created a fascinating graphic that charts the responses of 17 mattress companies and what they would disclose about their products. The full article, The Stuffing Dreams Are Made Of? and the graphic are must-reads for anyone looking to buy a mattress.