Yes. A spouse and unmarried children under 21 can obtain L-2 status, and L-2 spouses are generally employment-authorized incident to that status, independent of the specialist's own specialized-knowledge classification.
L-2 family benefits do not depend on the specialized-knowledge argument itself
Once L-1B status is approved based on the specialist's company-specific procedure knowledge, family members qualify for L-2 status through their relationship to the specialist, not through any separate evaluation of the underlying specialized-knowledge claim. A spouse on L-2 status is generally employment-authorized incident to that status, which is a meaningful practical benefit for a household relocating around a transfer that took real effort to document. The distinction between certification and company-specific knowledge that had to be argued carefully in the principal petition has no bearing on the family filing; what matters there is the relationship itself and keeping the L-2 applications current with the principal's approved status and any extensions.
Households that have moved between assignments usually hold a scattered set of documents across several countries, and the practical advice is to consolidate them before they are needed: each family member's passports including expired ones, birth certificates, school records and any prior immigration documents.