I think I have a more sensitive nose than most people, because I have to let a lot of things air out before I can stand to use them (like anything made of leather, and I don't use vinyl at all if I can avoid it), so this probably doesn't apply to most people, but for those few like me, here's my experience. I tried washing my mat with vinegar, and then I had it sit out in the garage for over six months, and I still couldn't stand the smell. I tried this with two different brands of mat, thinking maybe one was particularly bad, but they were both unusable for me.
I ended up buying a mat made from polypropylene (also familiar as recycling code 5) instead of PVC. Specifically, I bought the Fiskars Eco cutting mat. No smell. It's made of the same stuff a lot of plastic kitchen storage containers are made from.