Is the vomit of a child impure? And does it have to be removed from clothing for the prayer to be valid?

The correct view, and Allah knows best, is that vomit is not impure. Vomit.

It is not impure, it is not one of the impurities, so it is only cleaned as a matter of cleanliness and you can pray, it does not harm, inshaAllah, whether for adults or children.