Staphylococcus aureus can cause toxic shock syndrome. One symptom of toxic shock is vomiting. Hence, vomiting can be a sign of bacterial infection.
Vomiting can also be a sign of food poisoning, which also is caused by bacteria.
As far as presentations go, my knowledge of them is strictly limited to the descriptions in the medical literature--I am a researcher, not clinician.
The differential diagnosis for vomiting is huge...
Infection yes, but also headache, food poisoning ( as you mentioned) enteroviruses, motion sickness etc. Without waiting for the coroner’s report there is no way to know cause of death.
As a nurse who has given many vaccines the indication depends on whether the girl was febrile. Remember too that the general populations idea of what a fever is is different from the medical community. You are not febrile unless you have a fever of 100.1 or over.