Generally yes. An L-2 spouse is employment-authorized incident to status and does not need a separate application in order to work. A child's L-2 eligibility, by contrast, ends at twenty-one, whatever stage the assignment has reached.
One household, two different rules
Hypothetical example: Ferdinand transfers from East Ferris on L-1B for what the company calls a two-year assignment, his wife Reyna intends to work, and their son turns twenty-one in the eighteenth month. Reyna's route is the straightforward one, since employment authorization comes with L-2 status itself; the practical questions for her are professional licensing and finding the role, not permission. Their son's route is a deadline.
Unmarried children under twenty-one hold L-2 status, and reaching twenty-one ends it, so an alternative has to be in place before the birthday rather than discussed after it. A student route is the usual answer and needs an admission, evidence of funding and real lead time. Two related points are worth writing down early: the marriage must be legally recognized, since a common-law partnership is not a spousal relationship for this purpose, and L-1B is capped at five years, so the extension timetable should be mapped alongside the birthday.