No. Employment authorization comes from status, not from ownership, residence or payment. A spouse holding qualifying E-2 dependent status is generally employment-authorized incident to that status; check what was actually issued, including the I-94 and any E-2S notation. Unmarried children under 21 receive no work authorization from dependent status.
Confirm each person's own eligibility
Family members can require their own applications and appointments, so dates are not shared or guaranteed. Record each person's citizenship, marital or age status at the time of filing, and current U.S. status; a child approaching twenty-one changes the planning.
Separate the visa's validity dates from the period of stay recorded on the I-94, and keep the intention to depart when status ends part of the household discussion. A promise of a role in the family company is an employment arrangement; permission to hold that role is a status question, and shares held by a relative answer neither. Where a spouse or an adult child is expected to work in the business, identify the status each of them will actually hold and what it authorizes, then check the document issued to that person rather than the plan discussed at home.
The specific failure worth naming is a relative who begins working in the enterprise because the family owns it, without authorization of their own — a problem created by ownership being mistaken for permission. Ask before anyone starts, not after payroll has been run.