If a sister requests money from benefactors to give in charity to her needy relatives, would she be violating the prophetic tradition that advises against asking people, and is it permissible for her to use these funds to pay off her sister's debts, even if these debts are owed to the asker herself?

It is permissible for the sister to inform affluent individuals about needy people she knows and encourage them to give charity to them. This is considered a good intercession (shafa'ah hasanah), and she is not obliged to disclose the specific identities of the recipients to those giving the charity. She should give the charitable money to the needy sister so she can dispose of it as she wishes, and she should not take anything from it for herself, unless the needy sister gives her money after she has taken possession of it to repay her debt.