Yes. The spouse of an L-1 worker holds L-2 status and is employment-authorized incident to that status, so no separate work permit is required before accepting a job. Children under 21 hold L-2 status and may study.
L-2 spouses may work; L-2 children may not
This is one of the sharper contrasts between categories, and it often decides which route a household prefers. An L-2 spouse's authorization flows from the status itself, which means employment can begin without waiting for a separate document, although many employers and payroll systems will want evidence of status in hand. Children admitted as L-2 dependants may attend school at any level but hold no work authorization, and a child who turns 21 no longer qualifies as a derivative.
Because L-1A is capped at seven years and a new-office approval starts at one, a family should map the status of every member against that timetable early. A student nearing 21 may need an independent route, and that planning is easier before the first approval than during an extension.