This is an easy one to answer for you. The "Carpet and Rug Institute" (CRI) administers programs called "Green Label" and "Green Label Plus".
Click HERE for information regarding your carpet padding. You will need to look for padding that is certified by the CRI as having the "Green Label" (check out the logo on the page there.)
Remember also that the padding is put down with adhesives that could contain VOC's, so you want to also be aware that adhesives used are certified by the CRI as having the "Green Label Plus". Click HERE to find out more about that.
You want to assure that your carpeting also has the "Green Label Plus" certification.
Even with all this, especially if you are super sensitive, I would do an install at a time of year where you could ventilate your home for at least a week to clear our any potential off gassing. You will likely be installing products with low-toxicity ratings, not zero-toxicity.
Hope that helps. Good luck!