Yes, once the spouse holds L-2 status. Spouses of L-1 principals are employment-authorized incident to status and may work for any employer. Children in L-2 status may study but may not work, and each child's status ends at 21.
The spouse's authorization comes with the status, not the job
Hypothetical example: the manager's spouse is a horticultural technician who has an offer from an orchard near the subsidiary's yard. Nothing about the offer needs to be attached to the L-1A petition, because the spouse's permission to work does not depend on the employer. It depends on holding L-2 status as the spouse of an L-1 principal, which is documented on admission and reflected in the arrival record. The spouse should not begin work before that admission, and should keep the admission record because employers verify it. The orchard is free to hire the spouse in any role, including one unrelated to the principal's company. The couple's seventeen-year-old, admitted in L-2 status, can attend school but cannot take a summer job at the same orchard. When the child reaches 21 they must have their own status, so a family intending to stay several years should plan for that date rather than discover it.
Hypothetical example: the manager's spouse is a museum conservator who receives a contract from a regional museum after the family arrives. The first review should confirm the spouse's admission classification and retain the arrival record before the museum fixes a start date; the museum's payroll file, not the principal's petition, is what later documents the work arrangement. Separately, the family should calendar the principal's L-1 limit: qualifying employment generally requires one continuous year abroad within the preceding three years, and an L-1A principal faces a seven-year maximum. A child nearing 21 needs an independent-status plan rather than an assumption that the parent's extension will carry them forward.