Is it permissible for a divorced woman in Saudi Arabia who does not have a mahram, and whose ex-husband’s mother refuses her travel with her two children to her family's home, to remain in her ex-husband’s house for the sake of caring for her two children, knowing that her ex-husband’s brother is the one who takes them out for their needs?

There is no harm in you traveling with your children to your country and family if their father grants permission, and the paternal grandmother has no right to object. You are permitted to reside with a mahram, or in independent accommodation separate from non-relatives, or in a shared home with assurance against khalwa (private seclusion with a non-mahram). Your going to the market in the company of the children's grandmother is permissible. It is not permissible for you to leave them and travel away from them if it will harm them, unless their grandmother or someone else will manage their affairs and guarantee their rights. It is best to choose what is most beneficial for your children regarding residence or travel, and your travel should be with a mahram.