No. Funding and status are different questions. A qualifying E spouse may work incident to valid status with appropriate evidence; children lack that benefit solely as dependents.
Identify the actual role and category
Explain genuine ownership and contributions without assuming an account transfer grants immigration permission. The dependent category is for a qualifying spouse or unmarried child under 21; each person needs valid status.
The spouse of a principal E-2 investor is admitted in derivative E-2 status and is treated as authorized to work for any employer, including the enterprise, without needing a separate employment authorization document. For a household moving on a single business, the spouse's ability to earn elsewhere is often what makes the first two years survivable, and here it comes with the status rather than requiring a separate application. The spouse's authorization to work arises from the status itself rather than from a separate grant, so employment with an unrelated employer is available, subject to satisfying an employer's ordinary verification requirements. The principal must show treaty nationality, qualifying enterprise nationality, irrevocably committed at-risk funds, substantiality, a real nonmarginal operation, and development and direction through half ownership or operational control.