Yes. An L-2 spouse is employment-authorized incident to status after admission with the spouse annotation and may work for the family's own employer or any other. L-2 children are not employment-authorized.
Spouse work is permitted; the annotation makes it verifiable
L-2 spouses have been treated as employment-authorized incident to status since the policy change that followed litigation on the subject, and the authorization covers any lawful employment, including a retail role at the same company that employs the principal. On arrival, check that the admission record bears the L-2 spouse annotation; if it was omitted, request a correction before the first shift, because the dairy's employment verification relies on that record. The spouse's own nationality does not matter.
Children under 21 hold L-2 derivative status, may enrol in school and have no work authorization; a child reaching 21 during the assignment needs an independent basis to stay, which should be planned before departure rather than after the birthday. If the technician's status later changes to L-1A because he takes over production management, the family's L-2 status continues on the same terms.