I experienced a flight disruption during a business trip. My company paid for the tickets. Who should claim compensation and who will receive it?

You. As the passenger, you have to start the claim (here is the online form) and if it is successful, you will collect compensation.
The European regulation is here to protect individuals (passengers) and repair the wait, stress, tiredness they could have experienced.
This compensation is different from the reimbursement of tickets that can be asked by the employers to their travel companies.