Vomiting - Mouthful vomiting contains water or pus or blood or food invalidates the wudu, vomiting contains cough does not break the wudu. Touching the private parts with the bare hands (not according to Hanafi Madhhab).

How wudu is broken?

Wudu is broken or invalidated by discharging solid or liquid or wind from private parts including sexual intercourse, bleeding, vomiting, falling asleep or taking any intoxicating substance.

Does fart break wudu?

Unless you have a condition that makes it impossible for you to stop breaking wind long enough to pray, you'll need to renew your wudu whenever you nullify it and want to pray again. Inside farts will not break your wudu in any way because it is not physical.