I assume you're aware that plywood contains glue as well as MDF - just not as much? With regards to your son's condition, are you worried about the dust generated during construction, or is it off-gassing after the speaker cabinets are finished?
I would be most concerned about the dust, as opposed to off-gassing. When you think about it, off-gassing can't be avoided - just reduced. New furniture, carpet, draperies, paint, etc. Anything synthetic will off-gas to various degrees. So, a few speaker cabinets won't make a lot of difference.
I hope you have a dedicated woodworking area, separated from the living area of your house, like a shed or garage. My shop is in my basement, because I have no space on my property for a separate workshop. Dust is a big concern for me, so I've sealed up every little opening I can find and I have a dedicated dust collection system for my tools. Shop vacs can work, but they just don't pull a sufficient volume of air IMO. Plus, you need a HEPA filter cartridge for a shop vac, or you'll just be spreading the dust around more. Dust masks are uncomfortable, as Isiberian says, but I still use one - a proper one that seals to my face and has HEPA filter cartridges. On top of that, I have an ambient air dust collector. You cannot be too paranoid about woodworking dust - whether it's natural wood or MDF.
3/4" MDF is a lot cheaper than 3/4" cabinet grade plywood. However, such plywood can have your choice of veneers - oak, maple, etc. You just need to hide the edges. You can use iron-on veneer strips or mitre all your edges at 45 degrees, which would be a royal PITA.
If I were you, I'd stick with the MDF (make sure your son is not exposed to the dust). Assemble the cabinets with yellow carpenters glue, as it's much healthier than polyurethane glues or contact cement. It also creates a very
strong joint and provides air sealing for your cabinet, if you ensure a seamless glue join. As previously stated, screws don't hold well in MDF. You could also use a low-VOC paint to finish them. You stated that appearance isn't important, but plain MDF is butt-ugly.
Hope that helps. Well, I hope it doesn't hinder, at least...