What is the ruling on paying a sum of money to an employee to obtain a Qi Card, and what is the ruling on the aid received through it?

If you are unable to obtain the card through legal means, elevate the matter to the responsible authority. It is impermissible to pay a bribe when it is possible to complain to officials. However, if you cannot obtain the card except by paying, then there is no blame upon you, and the sin is on the one who takes [the payment], because you are unable to secure your right except through that means.