Is it possible to catch a stomach bug and not vomit?

Yes. Some 'stomach bugs' cause nausea and diarrhea (e.g. Norovirus, staph, bacillus cereus), while others cause mostly diarrhea (e.g. Shigella, salmonella, campylobacter, and some other viruses.).

Related Questions

Can you catch the stomach bug from someones nasal secretions or do you have to be exposed to vomit or feces. Vomiting started 24hrs after exposure?

Possibly, yes. "Stomach bug" usually means viral gastroenteritis, and the most common cause is norovirus. Noro is extremely contagious: just being in the same room with someone who vomits sometimes is enough to catch it, even if no direct contact. If somone with noro has cold symptoms, contact with nasal secretions probably could transmit it. OTOH, 24 hours is a bit soon: symptoms usually begin 36-60 hours. Read more...

Husband has a 24 hour stomach bug. Can I catch it if he breathes on me, hours after his last vomit, but I smell vomit on his breath? How to avoid it

Gastroenteritis. Without knowing exactly what has caused your husband's symptoms it is difficult to advise. Would not kiss him, not come in contact with his secretions from any end of his GI tract and try to stay reasonably far away. Not easy, but could be of many sources. Read more...
Gastroenteritis. does not have airborne transmission but intimate contact (or maybe droplets) could spread it. Read more...