Doesn't Elemental Designs use some sort of material for its cabinets that's supposed to be greener than MDF? I have no idea what it is, but it may be worth going to their site to find out what it is and to ask them where you might be able to get some.
And I don't know what it's impact would be on an asthmatic (I'm one, too, but I don't do woodworking).
It's gotta be cost-competitive with MDF as their subwoofer prices are extremely competitive. Granted, they buy a lot more than you or I might, but it's worth looking into.