I've had chronic asthma and allergies for as long as I've cooked. Usually if I need to cough or sneeze or whatever, I usually feel it coming at least a second or two before it happens. I take the opportunity to set down my knife or anything dangerous I have, and take evasive maneuvers. I try to stifle sneezes, but coughs are trickier. If I can get myself where I can ensure my cough is nowhere near food, all the better. Typically, I'll have to wash my hands, but most kitchens have hand sinks everywhere anyways, so it's not a big issue. If I sneeze or cough directly into my hands, I'll give them an extra thorough scrubbing, but it still usually isn't a lot of wasted time. I've seen many cooks with habits much worse than mine, and none of those restaurants have been shut down, or those cooks canned. Of course, I don't condone their unsanitary behaviour, but I merely want to say that in my experience, many cooks deal with wily bodily functions and are still able to hold down their jobs.
Short answer: you can be a chef if you have asthma. Just don't spew saliva into my food.