What is the ruling of Islamic law regarding obligating a wife to compensate a woman for goods she did not receive, given that the shopping websites were fraudulent, and the wife and her husband have lost a lot in attempts to provide compensation, yet the woman is still demanding the money?

It appears that your ruling is that of an agent, and an agent is not liable unless there is negligence or transgression. You are not obliged to guarantee the money as long as no negligence has occurred on your part. If you voluntarily guarantee it, this is a favor from you. We advise you to diligently follow up on the money you sent via the bank. You are permitted to take it, and if you are content with sending it to the lady, there is no harm. A condition for your work to be permissible is that the goods must be Islamically lawful.