How awful! I too feel sick when around smoke. The problem, I think, is that they pay rent and, as long as there is nothing explicitly stated in the rental agreement, they are probably allowed to smoke in their apartment. If it's really cold, I think they would not want to smoke outside.

First I would approach the neighbors, politely explain that the smoke makes you feel sick and wonder if there is any solution they have. If they are not reasonable, then there is nothing to be done. Second, I wonder if you could get a note from your doctor stating the impact this has had on your health. Perhaps you could use this (consult with a lawyer) to get out of your contract? Third, do you know what would happen if you broke your contract? One place I was in would ask for a months rent for leaving early. It seems like a waste of money, but if you can't function properly, it's money well spent.

I hope something works for you.