Yes. An L-2 spouse is employment-authorized incident to status, so no separate application is needed to be authorized. Practical steps such as obtaining a Social Security number and the employer's verification still take some time.
Authorization is immediate; the paperwork trail is not
Hypothetical example: a Mission dental hygienist accompanies her spouse, a transferred plant manager, on L-2 status and has a job offer waiting. Her authorization to work exists from admission because L-2 spousal status carries it, and her admission record will bear the spouse annotation that employers can accept. She still needs a Social Security number, and any state licensing for her profession is a separate matter that immigration status does not resolve.
L-2 children may study but may not work. Her status is tied to his: if his L-1A ends, so does her L-2 and her authorization with it. Couples planning on two incomes should note that the seven-year L-1A ceiling applies to the household's time in this arrangement.
Hypothetical example: A publishing-company executive transfers with a spouse who has a remote job offer and a teenager who wants a weekend retail job. The family review should treat their plans separately. The spouse's L-2S admission record is the key employment-authority document for I-9 purposes, while an L-2 child may attend school but cannot accept the retail work.
Check each passport expiration and I-94 expiry against the principal's authorized stay. If the teenager will turn twenty-one during the assignment, calendar the date early; dependant eligibility does not continue indefinitely simply because the principal's L-1A extension remains possible.