Because of the vomit of an 8-month-old baby, are all the places the baby crawls on impure, along with the mother's and the baby's clothes and everything the mother touches with her hand? And does everyone who puts this baby on their lap become impure if the vomit has soiled the baby's clothes?

It is obligatory to remove impurity, not to perform ghusl (definition) (ritual bath) for every prayer. Vomit is considered impure by the general body of scholars if it has changed from the state of food. If it has not changed, the Malikis hold that it is pure. So, if your daughter's vomit has not changed, it is pure. Even if it is considered impure by the majority, you can designate specific clothes and bedding for prayer, and wash whatever you are certain has been soiled. As for a small amount of vomit that is difficult to guard against, it is pardoned due to hardship, and this includes a baby's saliva and drool.