Hi FayH,
Yes it certainly is possible. I would also keep in mind though that all materials will have some “offgassing” or VOC’s (even the most natural ones will have some smell which is a VOC) but the real issue is whether the VOC’s are harmful or whether the smell is offensive to you (regardless of whether it is harmful or not). There are also many synthetic materials (such as polyfoam or memory foam) that have been tested and certified by CertiPur or other testing protocols such as Oeko-Tex to make sure that the amount and type of VOC’s would be “safe” for the large majority of people although most of these generally have some initial odor.
There is more in post #2 here about some of the materials that most people would focus on if they were looking for more natural materials or to minimize the risk of harmful VOC’s in their mattress and post #2 here and the other posts and sources of information that it links to has much more information for those that want to do more research into the complex issues that can help them answer “how safe is safe enough for me?”.
To bypass all the marketing claims that you will encounter in the market about foam materials I would focus on the certifications that they have. If you are sensitive I would also make sure than any cotton in your mattress is organic (cotton is one of the most heavily sprayed crops in the world) and in the case of wool I would focus on wool that is either natural or organic and that doesn’t use any harsh chemicals in the raising of the sheep or the processing of the wool. There is also more about organic certifications in post #2 here and the posts it links to.
I would also focus on making sure you can find out the specifics of the materials in your mattress which are much more important than the name of the manufacturer on the label.
Phoenix