Yes, once in L-2 status. An L-2 spouse is employment-authorized incident to status, which does not depend on whether the principal holds L-1A or L-1B. Children in L-2 may study but not work.
Practical points for the household
The spouse's authorization follows from L-2 status itself, so the employer in the affiliate's town does not need to wait for a separate approval, though the spouse should hold documentation that evidences the status and should check what a specific employer's onboarding process requires. The spouse may work for any employer, including one unrelated to the fabricator. Children may attend school in L-2 status and, as with every derivative category, a child who reaches 21 during the assignment needs a new basis to remain.
If the family intends to stay for the whole assignment, list each person's passport and Canadian status, since a spouse or child who is a permanent resident rather than a citizen will process L-2 at a consulate rather than at the border. A technician on a short installation assignment may choose to travel alone and have the family visit as visitors; that is a legitimate plan, but visitors may not work or enrol in a full course of study.