Yes. An E-2 spouse is employment-authorized incident to status and may work for any employer, including the investor's enterprise. Children under 21 in E-2 dependant status may study but are not work-authorized.
How to sequence the household
Sequence the spouse's plans after the investor's visa but plan them from the start. Hypothetical example: a Lévis dental hygienist's husband buys a U.S. landscaping business on E-2; she intends to work in the business office initially and later seek clinical work.
Her E-2 dependant status carries employment authorization incident to status, so office work for the enterprise is permitted, and outside employment is permitted too, subject to any state licensing her profession requires, which is a separate matter from immigration. Their seventeen-year-old may attend school in dependant status but may not take a paid job on that basis, and at 21 will need an independent status. Note that the spouse's nationality does not need to be a treaty nationality; only the investor's and the counted owners' do.
Gather marriage and birth certificates now, and record each child's birth date against the twenty-first birthday before choosing a visa validity period.