D.W.
I'd ask the pediatrician for a referral to a pediatric Gastroenterologist if you believe there's a chronic issue.
It's completely possible that it is a virus that she keeps getting reinfected with, but I believe that's highly unlikely.
Bilous vomit is not too uncommon. If there's nothing in the stomach, all that will be vomited will be bile.
If it were my child, I'd personally be pressing for more answers to find out what's going on.
Good luck - hoping she's getting better soon.