If my child is vomiting and has diarrhea, does he need antibiotics?

No. Probably not-- most causes of vomiting and diarrhea, at least in the developed world, are not bacterial infections and do not require antibiotics. In fact, antibiotics may make vomiting and diarrhea worse.
No. Vomiting is alarming, and annoying, but vomiting and diarrhea are nearly always virally related. We call it "stomach flu" but generally a gastroenteritis is a viral illness that cannot be cured by antibiotics. In fact, antibiotics kill the normal bacteria found in the intestine, and can intensify both vomiting and diarrhea. Best thing is to give fluids and watch for dehydration.