Sulfites are a natural product of fermentation, so you won't be able to find wine completely free from sulfites. US labelling requires the sulfite warning (or did 20 years ago, the last time I was in the US ;-) ), even if no more sulfites are added in processing.
Sulfites are often added at bottling time in order to reduce oxidation of the wine. If the wine is very young, it is usually filtered, which can introduce oxygen into the process. Thus young wines often have sulfites added. If a wine has been aged for a couple of years, it will not need to be filtered and so it might not have sulfites added. Depending on the winery, you may find that higher quality wines will have little enough sulfites that you won't get a headache.
Keep in mind my original sentence, though, if you are allergic to sulfites. No wine (or beer) is free of it, so if you are in danger, do not consume it.
