Is what the questioner and the travel agency are doing—exchanging the cash travel tickets provided by the company to the questioner—permissible, taking into account that the company does not object and the travel agency obtains discounts from airlines?

There is no objection to benefiting from the value of a travel ticket if the employer permits it and does not stipulate travel as a condition for its entitlement. However, defrauding the employer to take the ticket value without traveling is not permissible if travel is a stipulated condition. Nor is it permissible for a travel agency to collude in this if it is aware of the prohibition, in accordance with the Almighty's saying: "And do not cooperate in sin and aggression."