Scholars disagree on this, and it's a good illustration of how fiqh differences arise from how a text is read. The Hanafi school holds that vomiting a mouthful — enough that it cannot be held back without effort — breaks wudu, reasoning that impurity exiting the body in a notable quantity nullifies purity, similar to other discharges. The majority — Maliki, Shafi'i, and Hanbali scholars — hold that vomiting does not break wudu at all, since no hadith specifically lists vomiting among the things that nullify it, and the general fiqh principle is that a confirmed wudu isn't broken without clear textual evidence (the base list of nullifiers is built from situations like those referenced in Quran 5:6). In practice, a Muslim should follow whichever school or guidance they normally rely on; if unsure, renewing wudu after vomiting is a safe, precautionary choice and not wrong to do, even where not strictly required. Rinsing the mouth afterward is recommended regardless, for basic hygiene before resuming worship.
Q&A · Purification
Does vomiting break your wudu?
References
Informational, not a personal fatwa. Consult a qualified scholar for rulings on your situation.