Is it permissible to pay money to someone in order to transport my sister, the teacher, to Riyadh?

Bribery is one of the major sins, and its perpetrator is cursed, as the Prophet, peace and blessings be upon him, said: "The Messenger of Allah, peace and blessings be upon him, cursed the one who gives and the one who takes a bribe." And the 'ra'ish' (the intermediary) between them is likewise cursed. It is forbidden to request a bribe, to accept it, or to act as an intermediary between them.

As for the bribe through which a person obtains their right or repels an injustice, it is permissible according to the majority of scholars. In this case, the sin falls upon the recipient of the bribe, not the one who gives it. So, if you are compelled to pay money to transfer your sister after you have exhausted your efforts, and the transfer is not at the expense of another woman's entitlement, then there is no sin upon you, Allah willing, and the sin is upon the one who took the money.