。
She got bus-sick and started throwing up.
Literal
She [topic-は] bus [from-に] got-sick [and-て] vomiting-started.
酔う covers two related senses: motion sickness (車酔い, 船酔い, バス酔い) and intoxication (酒に酔う 'get drunk on alcohol'). The に here marks the cause/source — what made her sick. 吐き始めた combines the verb stem 吐き ('vomit') with 始める ('begin') — the productive 'start V-ing' pattern: 'began vomiting.' Bus motion sickness is common enough in Japan that pharmacies stock 酔い止め (motion-sickness pills) prominently, and Japanese travelers often use them preemptively for long bus rides through mountainous regions.