What is the ruling of Islamic law on a poor family using the sponsorship money of their orphaned sister to cover their household expenses, knowing that the sister agrees to this and a portion of the sponsorship has been allocated for her?

It is permissible to take from the wealth of a mature, sponsored sister to the extent she permits, and maturity here means good management of money. However, if she is not mature, it is not permissible to take anything from her wealth, even if she gives permission. If the sponsor sent the amount in her name and explicitly stated that it is for all of you, then it is permissible to take from it.