No. A child admitted as a derivative may attend school but is not employment-authorized on that basis, and that does not change because the employer is the family's own company. Only the spouse is employment-authorized incident to L-2 status.
Working for the family firm is still working
Hypothetical example: the engineer's seventeen-year-old has spent summers sweeping the Picture Butte shop and expects to do the same at the American plant. The relationship to the employer makes no difference: employment requires a lawful basis, and derivative status for a child does not provide one, whether the work is paid, unpaid, casual or seasonal. School attendance is permitted.
The spouse's position is different, since employment authorization comes with L-2 status itself. Derivative eligibility ends at twenty-one, so a seventeen-year-old and a five-year assignment should be looked at together. There are also child labour and safety rules in an industrial setting that apply independently of immigration status, and a processing or fabrication floor is exactly where they bite.