Yes. An L-2 spouse is employment-authorized incident to status and may work for any employer, including the family's own company, once admitted with the spouse annotation on the I-94. A commercial driving licence valid in the state is a separate matter.
Immigration authorization and the driving licence are two different checks
The L-2 spouse of an L-1A principal is treated as employment-authorized on admission, and that authorization is not tied to an employer, so driving for the U.S. subsidiary is permitted. Confirm the I-94 shows the L-2 spouse annotation and correct it before starting work if it does not, since the terminal's payroll verification will rely on it.
The immigration authorization does not replace state licensing: a commercial driver's licence must be obtained from the state under its own rules, and a Québec licence may or may not be exchangeable. Children in L-2 status may attend school but may not work at any age, and a child who turns 21 loses derivative status. The spouse's own nationality does not affect L-2 eligibility, unlike TD, which also does not restrict nationality but forbids work entirely.
Keep each family member's passport valid for the whole assignment.