You have used the term "agency" and "company" interchangeably throughout this thread, and yet they are very different things, so it's hard to know which of the two you are talking about, though I suspect based on the need for a car that it's not in fact an agency.
A cleaning "agency" does not employ staff. They are exactly as the name tells us - an agency. The client pays a monthly fee to the agency in exchange for an introduction to a self-employed cleaner, who the client pays directly. The client also provides all the cleaning equipment and materials, and it's up to the client to mostly manage the cleaner & work with the cleaner to arrange times to clean etc. It's irrelevant to the agency how the cleaner gets to the client, and when.
A cleaning "company" does all of the above. And yes, it very much DOES matter to them as to when the cleaner gets to the client and how they get there, as the cleaning company is providing a fully managed service, usually with equipment and supplies included. The cleaners may or may not be on a payroll, though the more successful companies will have them on the payroll. In this case it is essential a member of staff has a car - you can't say it's wrong, because it's what is needed for the job! Payroll staff have to be paid for all their travel time, at least they have to be paid enough each day so that their rate of pay does not fall below minimum wage, when hours worked and travel time is added together (as do self-employed staff but don't get me started on that point as I will never stop, because sadly they frequently aren't). To that end, transportation is essential.
However, in all of this, I don't understand why your cleaner is wanting to do anything but work for herself, as clearly if she has you as a client then she's more than capable of finding and managing her own...similarly, if she's that good, aren't you worried that she will leave you if she gets more than enough work from a cleaning company?