If your baby vomits full mouth, it is impure with nijasat ghalizah, it will be washed with water if it touches the cloth of the body:
قال في الدر المختار: و ینقضہ قئ ملأ فاہ ……… وھو نجس مغلظ ولو من صبي ساعۃ ارتضاعہ ھو الصحیح (در مختار مع الشامي: 1/265-266)
The cloth should be cleaned by washing with water thrice after wrenching each time, while the body should cleaned by wash away the impurity from the place of impurity until you are sure that it was removed.
If the vomiting is not full mouth, then it is not impure (napaak). In this case, it is not necessary to wash the body or cloth for taharah.
Allah knows Best!
Darul Uloom Deoband