Is it forbidden to take the basket designated for a female questioner in her name, and must she give it to the family that was unable to attend to receive it, especially after the supervisor refused to give the basket to them?

If you are eligible for the food basket according to the donors' conditions, then there is no harm in taking it. If you are not eligible, then it is not permissible for you to take it. You are not required to give the basket to the woman you promised it to, as long as the manager has not approved the provision of her basket.