No. Each person needs the relevant status and permission. A qualifying L spouse may work incident to valid status with appropriate evidence; children do not gain that benefit solely as dependents.
Distinguish a proposal from admission
A company’s plan or a principal’s filing does not automatically change relatives’ existing status. Review the individual records. The dependent framework generally covers a spouse and unmarried children under 21; each person must obtain and maintain the appropriate status.
Because the authorization arises from the status itself, a spouse can accept employment on arrival rather than waiting out an application; obtaining a document that evidences the authorization simply makes hiring easier in practice. Documentation practice still matters: employers completing employment eligibility verification will want to see the admission record, and it is sensible to carry the marriage certificate and the principal's approval notice. Children admitted as dependants may attend school at any level but may not accept employment, and derivative status ends at twenty-one, which for a family with a teenager is a date to plan around rather than a footnote.