It's fair to presume that the miles are yours to keep. I would not ask the company or even mention it to them. I'd just call the airline to attach my FF number to the reservation. If the airline says the person who booked the travel needs to do it, only then would I call the company's travel coordinator.

Be aware, though, that there are two cases where you might not get to accrue miles:

1) Depending on the airline or the method of payment the company used, it's possible that no FF miles will be paid.

2) Some companies and some government organizations have policies that employees may not keep miles. If you learn that this employer has such a policy and your job responsibilities would include periodic travel consider that along with other factors about the working environment.