Yes. An E-2 spouse is employment-authorized incident to status and may work for any employer, not only the family business. Children in E-2 dependant status may study but are not employment-authorized.
Spousal work permission is broad; children's is absent
Hypothetical example: the investor's spouse is a laboratory technologist who would rather work at a local clinic than in the cleaning business. Once admitted as an E-2 dependant spouse, she is employment-authorized incident to status and may accept employment with any employer; her admission record can serve as evidence for the employer's verification. Any professional licence the clinic requires is a state matter separate from immigration permission. The couple's twenty-year-old child may accompany them and study, but E-2 dependant status does not authorize the child to work, and dependant eligibility ends at 21. Plan for that date rather than assuming a renewal will cover it. Each person's citizenship should be confirmed separately; a spouse who is a Canadian permanent resident but a citizen of another country is assessed on that country's treaty status for a principal application, though as a dependant the spouse's nationality does not need to match. E-2 offers no direct path to permanent residence for anyone in the family.
Hypothetical example: a language-school proprietor brings a spouse and a seventeen-year-old child, while the spouse expects to work at an unrelated employer. A qualifying E-2 spouse has work authorization that comes with valid E-2 status; keep the status evidence available for the employer's verification process. The child may attend school but cannot work as an E-2 dependent, and eligibility ends when the child marries or reaches 21. Create a family calendar with passport validity, school dates, and the child's age milestones. The decisive records are civil documents and admission records, not the investor's business plan. If the child expects paid work, consider an independent option before the family relies on dependent status.