No. Children admitted as derivatives may attend school but are not employment-authorized on that basis. Only the spouse is employment-authorized incident to L-2 status, and derivative eligibility for children ends at twenty-one.
One derivative status, two different sets of rights
Hypothetical example: the specialist's household includes a spouse who wants to work and an eighteen-year-old planning to study and take weekend shifts. The spouse's position is settled by the status itself. The eighteen-year-old may attend school but would need an independent lawful basis for any job, such as a student route with its own restrictions on employment, and that is a separate application rather than an extension of the family's status.
Age matters twice here: derivative eligibility ends at twenty-one, and a five-year assignment started now would run past that point. Confirm each person's citizenship separately, since a family member who holds Canadian permanent residence rather than citizenship may face different procedural requirements than the principal does.