Is distributing sweets on the day of the Prophet's birthday—may God's peace and blessings be upon him—considered fulfilling the vow made by the questioner, or is it permissible to change the day of fulfilling the vow? And is it permissible for her and her children to eat from these sweets?

If you did not utter what you vowed, but it was merely an internal thought, then you are not obliged to fulfill it. However, if you did utter it, then it is a binding vow that must be fulfilled. Choosing the day of the Mawlid to fulfill the vow is permissible if it is not intended for its own sake or as a celebration. However, it is preferable to choose another day to avoid imitating those who innovate in religion. Your eating from it depends on your intention: if you intended to distribute all of it to the poor, then neither you nor your children may eat from it. If the vow did not specify a particular recipient, then there is no objection to you eating from it.