No. A child in E-2 dependent status may study but is not authorized to work. Only the E-2 spouse holds employment authorization incident to status.
The dependent child is the one who cannot work
The classification separates the family sharply. The investor works in the enterprise; the spouse is employment-authorized incident to status and may work anywhere, including elsewhere in the business or for an unrelated employer; a child under twenty-one may live in the United States and attend school but may not take a job, including a summer job in the family company. Working anyway puts the child's status at risk and reflects badly on the investor's file. The realistic options are to have the child work in Canada, to arrange an unpaid arrangement only where United States labour law actually permits one, or to look at whether the child qualifies for a status of their own such as a student classification with the work permission attached to it. Note also that the child ages out at twenty-one and needs a plan before then.
Hypothetical example: a seventeen-year-old child wants to handle weekend social-media posts for the family's U.S. business. The first review should identify whether the activity is productive work, who directs it and whether compensation is planned. The child’s admission category settles that no employment is permitted, even if the role is unpaid or described as helping family. Put another authorized worker in charge of the account and keep the child focused on school or non-work family life. Waiting until a compliance question arises is too late to establish a careful plan.