What is the legitimate Islamic formula for an agreement with a tourism agency to pay the costs of Umrah for a group of people, provided that the cost is paid in installments by them with a profit margin?

The validity of an ijarah (leasing/renting contract) requires both the rent and the usufruct (benefit) to be known, in order to prevent disputes. In the described scenario, the ijarah was made for usufructs that are fi al-dhimma (in obligation/liability) and for installments, which is a point of contention among scholars.

It is permissible for you to agree with an umrah performer on specific specifications and services, and then contract with tourism companies for the same specifications and services. This is called "parallel ijarah" and there is no harm in it.

It is not permissible for someone who has rented a specific item (ayn mawsufah fi al-dhimma) to re-rent it to another person until after taking possession of it. However, they are entitled to rent out an item described fi al-dhimma with the same specifications, without linking it to what they themselves rented, and then deliver it to the second renter after taking possession of it.