You don’t necessarily need to have a formal policy, but if it is informal it should be consistent across the board. It’s best not to offer it to just certain people or departments, especially if you’re unable to accommodate having an entire department work from home. In those cases, perhaps you offer flex-time. If people stay late some days and want to head out early on a Friday, let them.
In many cases, an employer has to meet compliance regulations and this is why you are being asked for certain information. Even if you work with a professional recruitment firm and are being submitted for a direct hire position you will have to go through both the process for the recruitment firm and the employer.