Yes. An L-2 spouse is employment-authorized incident to status, so employment does not depend on a separate grant. He should still confirm any occupational licensing his particular job requires in that state.
L-2 employment comes with the status
Hypothetical example: the manager's husband drives a school bus in the Grand Bay-Westfield area and assumes he will be idle for the first year. He will not be, at least as a matter of immigration status: spouses in L-2 status are authorized to work incident to that status, which is a different and far simpler position than the one TD dependants are in. What he does need to check is the non-immigration side.
Commercial driving licences do not transfer automatically, states set their own endorsement, medical certificate and residency rules, and school employers add background screening on top. He should start that enquiry with the state licensing agency before the move rather than after, because the practical delay in his case will come from the licence, not from his status. Their children would hold L-2 status too, and may attend school, but may not be employed.