There are really two issues at play here; one is the VOC offgassing and the other is your mold and mildew problem.
VOCs, or volatile organic compounds, are most prevalent in carpet, vinyl and laminate flooring products, so stay away from these. They have the capacity to offgas for years after installation so this can be a serious issue, especially in a tightly sealed house in the winter when you have all of the doors and windows closed.
Your best green flooring options are hardwoods, bamboos, cork, tile or other hard-surface finishes like slate. Hardwoods, bamboos and cork can all be purchased pre-finished with either very low or no-VOC finishes; just check the manufacturer's information.
These products can also typically be laid with a glueless application so as to avoid the VOCs associated with adhesives ("floating floor" type products). The other advantage to any of these hard surfaces as opposed to, say, carpeting is that they will hold up much better to any future moisture issues like the ones that have caused your mold and mildew problems.
The next culprit to look at is your subfloor, which you have indicated is made out of plywood. Plywood, and similar products like MDF or particle board, generally use formaldehyde-based binders and can offgas formaldehyde for many years after installation. Formaldehyde is a known carcinogen and something you do not want to be exposing your family to.
You can investigate sealant/primer products to coat your subfloor which will significantly reduce the offgassing of formaldehyde from your existing plywood subfloor if you do not want to replace it.
- AFM Safecoat makes several no-VOC primer type products.
- For wonderful no/very-low VOC floor sealers, look at Bona Products; they make some great high-traffic products -- however, they do come at a price premium.
Whatever product you choose, just make sure that you work closely with your installer. The VOC content in any flooring application is going to come primarily from either adhesives or finishes, which much of the time are controlled by the trade professional doing the installation.
For more information:
Read Mari Strain's Ask A Pro Q&A "Can you recommend healthy flooring for a home with kids and pets?"
Also, architect Ian MacLeod offers detailed advice about subflooring in his Q&A "I'm concerned about toxic offgassing from OSB subflooring and roofing. Are safer products available?"