I'd like to know this also. A surprisingly difficult question to answer unless you have a lot of money to spend.
If you have several thousand dollars to spend, there are certified organic mattresses which should be safe. Also there is a mattress company that was supposedly started by someone with MCS (but who knows really) called My Green Mattress. They have mattresses for slightly under $1000. Since I do not have that kind of money....I can't say how good but I will say: don't buy what I bought! (Read on only if you're considering memory foam or if you have concerns about toxic flame retardants).
This has been a huge challenge in my life as I have recently moved and ditched our mattresses due to mold in house.
I purchased memory foam mattresses (the kind sold on Am@zon and shipped rolled up in a box) due to affordable price (resources exhausted due to living with undiagnosed mold illness for years). Did research and knew that memory foam has a chemical odor but most reports are that the odor goes away when aired out properly, which I did.
In my opinion and in my experience, people with MCS should avoid these memory foam at all costs! I have been made sick by them (migraines, muscle weakness, etc) and even my family members, who up until now were not particularly sensitive, are also getting sick. And don't be fooled by the Certi-Pur foam certification which sounds good and talks about VOC levels but really is just marketing bull. Don't be fooled either by other marketing ploys that imply that the mattress is safe such as "charcoal infused" or "green tea infused".
The off-gassing from foams and adhesives is one thing but when you also consider the chemicals used to make mattresses flame resistant, the problem becomes very complex! I thought I would just replace my toxic mattresses with some "100% cotton" futon style mattresses and live with the back pain until I figure something out. Foam-free and I can't detect an odor, yes, but then I realized that I have no idea what the manufacturer uses to achieve the flame retardant standard. Those chemicals are typically endocrine disruptors and my wife was just diagnosed with Hashimoto's. Neither the retailer or the manufacturer will comment or answer my simple question, nor are they required to.
You asked a great question and I will follow this thread closely for new information. Good luck Kimberly.