will the USA Embassy refuse to grant visa to my Daughter and granddaughter?

A seven-year-old child is not barred from obtaining permanent residence by virtue of overstaying in the US. She would have to be 18 to be subject to a three or ten year bar for being in the US illegally for 180 days or one year respectively. As to whether the US Embassy would or would not refuse the visa for other reasons is up to the interviewing consular officer. I do note that if a consular officer makes a finding that the parent intentionally intended for the child to overstay at the time that she applied for her visa, such could constitute a reason to deny. However, such a denial may be overcome through a waiver.Due to the limitations of the Lawyers.com Forums, Alan Lee, Esq.'s (the "Firm") participation in responding to questions posted herein does not constitute legal advice, nor legal representation of the person or entity posting a question. No Attorney/Client relationship is or shall be construed to be created hereby. The information provided herein by the Firm is general, and requires that the poster obtain specific legal advice from an attorney. The poster shall not rely upon the information provided herein as legal advice nor as the basis for making any decisions of legal consequence.