Yes. An L-2 spouse is employment-authorized incident to status, with the annotated Form I-94 serving as evidence, so he may work for any employer or for himself without a separate permit. Children in L-2 status may attend school but may not work.
The spouse's freedom is broad, and it ends when hers does
Among the routes described on this site, L-1 treats an accompanying spouse most generously. Delphine's husband could take employment anywhere, work remotely for a Canadian employer or start his own business, relying on the annotated I-94 that employers ask for during onboarding. Unmarried children under 21 enter in L-2 status, may enrol in school and may not work.
Three limits deserve attention. The family's status is derivative: if the acquisition is unwound, or she leaves the group, the L-2 statuses end and any job her husband holds must stop. A child approaching 21 needs an independent plan before the birthday.
The marriage must be legally valid, since a common-law partner does not qualify. His own nationality is irrelevant to L-2, so Canadian permanent residence rather than citizenship changes nothing except whether he needs a visa stamp.