Is it permissible for an employee to request a sum of money for personal purchases, in exchange for giving the merchant a fictitious purchase order, through which the merchant recovers the amount from the administration's treasury?

Bribery is one of the major sins, as stated in the Hadith (definition): "The curse of Allah is upon the briber and the bribed." An exception to this is paying money to obtain a right that cannot be acquired otherwise. In such a case, it is permissible for the payer to give the money, while the receiver remains a briber. The scenario you described is not bribery; rather, it is embezzlement in which you are a partner and an accomplice. You bear its sin and punishment because you assisted the employee in his crime. Allah Almighty says: (And cooperate in righteousness and piety, but do not cooperate in sin and aggression).