No. Children admitted as derivatives of an E-2 investor may attend school but are not authorised to work in the United States, including in the family's own business. The investor's spouse is in a different position, being employment-authorised incident to that status.
One household, three different permissions
Families running a small business together assume the household shares one permission, and it does not. Three distinct answers apply under the same roof. The investor develops and directs the enterprise.
The spouse holds employment authorisation incident to E status and may work, for the family business or for anyone else. Children under twenty-one accompany the household, may study, and may not take employment, and paying a child for stall work in the family barn is employment whether or not anyone calls it that. Hypothetical example: a family from Saint-Joachim-de-Shefford runs a lesson barn where a sixteen-year-old teaches beginner classes on Saturdays and is paid from the till.
That arrangement is not permitted by the child's status regardless of the amount involved. Two further limits belong in the same conversation. A child ages out of derivative status at twenty-one and needs a route of their own from that birthday, most often a student status arranged well in advance.
And E-2 leads nowhere by itself: no length of stay converts it into permanent residence for the investor or for anyone in the household.