The German verb for “to vomit” or “to throw up” is either “erbrechen” or the reflexive “sich übergeben”, which literally means “to give oneself over”.
Watch out though, the noun for vomit is NOT “Übergabe” (f.), which means “handover”. You‘d rather use “das Erbrochene” in polite speech.
Other more polite terms are “brechen” and “spucken”, although these words are ambiguous as they mean “to break” and “to spit”, respectively.
Vulgar, but very common expressions (similar to “to barf” or “to puke”) are “kotzen” (with the noun “Kotze” referring to the resulting product), “reihern” (from the bird “Reiher”, which is a “heron” in English, and regurgitates food for its young) and, more rarely “kübeln” (from “Kübel” for “bucket” or “tub”) or the sophisticated “(aus)speien” (“to spew” or “to spout”).
The medical term is “vomieren”.
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Examples
Fritz hat sich die ganze Nacht übergeben. Fritz threw up all night.
Pass auf, da ist Kotze auf der Straße. Watch out. There’s puke on the street.