Is it permissible for a divorced woman with children, whose father is alive but does not provide for them, to state in a housing contract that she is a widow and her children are orphans in order to obtain housing from the Zakat Fund, given her dire need?

The father of the children must be sought out and compelled to provide for them. If that is not possible, then it is permissible to give them zakat (definition). It is not permissible for the woman to lie and claim she is a widow in order to receive zakat. Rather, she must be truthful in stating her situation to the relevant authorities, and her needs will be met without the necessity of lying.