Is it permissible to go for Umrah with a three-month-old baby girl and her mother who recently underwent a Caesarean section?

The statement, "We implored Allah Almighty that if He blesses us with a newborn, male or female, we would take them for 'Umrah," does not carry the ruling of a vow (nadhr) unless it includes wording that indicates an obligation to perform an act of obedience. However, it is preferable to fulfill what was intended. If you utter words that indicate an obligation to perform an act of worship, then it is a conditional vow (nadhr mu'allaq) which is disliked, but it must be fulfilled when the condition upon which it was made occurs. It is advisable to hasten in performing the 'Umrah to resolve any difference of opinion. The 'Umrah of an un-discerning child is valid and you will be rewarded for it. There is no harm in delaying the 'Umrah if there is an impediment preventing the journey.