This just ran through our house. My rule- nothing by mouth until vomiting has stopped for at least an hour if not longer. Then a sip or two every 10 minutes of sparkling cider, Sprite, really whatever appeals to the child. If they hold that down, I'll let them have some bland food 6 or more hours after last vomiting. Usually they don't want to eat anything for a day or two. No worries as long as they're drinking.
I wouldn't give Zofran, nor would I go to urgent care for a stomach virus. You want him to vomit- it's how the body eliminates the virus. For my kids, the vomiting phase usually doesn't last longer than 6-12 hours. Then they're lethargic and puny for another day or two and then it's business as usual. Only once did we have to go to the dr for a stomach bug and that was when we all got norovirus. My oldest daughter couldn't get past the vomiting part after a week and needed meds. That was more extreme, but even still she just took some reglan and was fine.
Hope he feels better soon!