When is my Employee DRIVING ON-THE-JOB?
In a home care situation on-the-job driving typically includes the following:

1. Driving from one client's home to another, without significant deviation.
2. Driving a client or driving on behalf of a client.
3. Driving for any other business purpose (picking up office supplies, going to the bank)

So far, the commute (from home to the first visit, or from home to the office) is NOT considered on-the-job.  FindLaw has some interesting information HERE on some of the ongoing issues in this area.
What Happens in the Event of an Accident?
Let's walk through a claim scenario and see what coverage comes into play for each injured participant:

A caregiver (in her own car) is driving a client to a doctor's appointment and rear-ends a car at an intersection.  The client suffers a broken pelvis and significant bruising. The occupant of the other vehicle has a severe neck injury and is taken to the hospital. The caregiver suffers a broken collar bone.

1. The easiest coverage to determine is workers comp.  Workers compensation always       covers on-the-job injury to an employee.  The caregiver will receive 100% of medical       bills paid and 70% wage replacement.      

2. Damage to the client.  Typically a severely injured client will seek damages from a           home care agency by alleging negligent driving.  Most general liability policies                 exclude auto accidents as a covered event. An agency must add non-owned auto         liability as a rider for coverage to apply.

3. Damage to the occupant of the other vehicle:  The 3rd party to the accident will most       likely file claims with both their health insurance and auto insurance companies. Both       of these companies will then initiate subrogation actions against the caregiver's               personal auto insurance and the home care agency.  The personal auto policy is             always primary     (first in line) and after that, the agency's non-owned auto coverage.
What Insurance Fixes My Employee's Car?
Not yours!  There is not an insurer crazy enough to extend personal auto coverage to all of your employees. Non-owned auto provides liability protection for your agency but does not provide physical damage coverage to the caregiver's car. Please consider adding wording to your employee handbook that states that all employees should make sure their personal insurance will cover their risk for driving on the job.