Often firms and institutions rent out apartments for or on behalf of their employees. These are usually for a month or so (e.g. if the employees are in Berlin for a project). Alternatively, the employees need a furnished place for a couple of months until they have an apartment of their own in Berlin.
It is nomally easier to complete the Rental Agreement with the direct tenant rather than the firm even if the firm is paying for it. This way you have a real person with whom you can get in touch directly and who is responsible for the agreements made in the contract.
If the firm does not have an office in Germany, then you should always complete the discussions and contract with the actual tenant and not the firm. If you rent out to a company, then they should state in the rental agreement that they cannot transfer the apartment to several different people within the same rental period.
It is usually better to have the rental agreement concluded with the tenant directly. If it is a very big corporation that routinely deals with such temporary renting out for employees, then you can send the rental agreement to the firm directly.