Is it common for an one year old to vomit? Usually drank too much milk? I see that my child puts her hand in the mouth and it may induce vomit? She has no fever, is it something to worry?

Your child must have a sensitive gag reflex. You need to fix this right away because you don't want your child to have a negative reaction toward feeding and eating in the long run, since gagging is an unpleasant experience.