No separate work permit is needed: an L-2 spouse is employment-authorized incident to status. What the spouse should start early is any state licence or credential review the intended occupation requires, because that runs on the state's timetable.
Skip the permit, start the licence
Hypothetical example: the specialist's wife is a pharmacy technician who assumes she must obtain an employment authorization document before she can be hired. As an L-2 spouse she is employment-authorized incident to status once admitted, and her admission record can serve as evidence for the employer. What will take time is the state pharmacy board's certification or registration process, which may involve an examination and a background check; she should begin that as soon as the petition is filed.
The family's sixteen-year-old may attend school in L-2 status but is not employment-authorized on the basis of that status; any later work would need its own basis, and dependant eligibility ends at 21. Verify each family member's citizenship; a spouse who is a Canadian permanent resident rather than a citizen may need a visa stamp before travel, which should be sequenced after the petition approval. The family review should separately confirm the principal's prior L time, the spouse's status evidence, and every child's age and school or travel needs.
It should also reserve time for documents issued outside the country of residence. Hypothetical example: a marine-inspection specialist plans a move with a spouse whose passport name differs from the marriage certificate and a child attending a boarding school. The reviewer orders the name-link record, confirms travel consent needs with the school, and keeps the dependent records separate from the specialized-knowledge evidence.
A simple relationship discrepancy can otherwise delay a case whose business evidence is ready.