Not as an E-2 dependant. Children may accompany until twenty-one and may study, but they are not employment-authorized. He would need his own status permitting work, and once he turns twenty-one his dependant status ends regardless.
Dependent children may study, not work, and age out at twenty-one
The E-2 dependant rules distinguish spouses from children. The spouse of a treaty investor is employment-authorized incident to status and may work in the inn or anywhere else. A child, by contrast, may be admitted as a dependant only while unmarried and under twenty-one, may attend school, and has no employment authorization, so a twenty-year-old cannot lawfully take shifts in the restaurant or on the front desk even unpaid, since unpaid work for the family business is still employment for this purpose.
His dependant status will end on his twenty-first birthday, at which point he must hold another status or leave. Options include student status if he is admitted to a U.S. program, or, later, a category of his own if he qualifies.
For the Harrison Hot Springs couple, staffing the inn should assume the son is not part of the workforce, which also matters for the marginality analysis, because jobs filled by family members do not demonstrate capacity beyond the household. The first review should create an age-and-status calendar for every dependant, including passport expiry, current I-94 information, school plans, and any separate employment goal. Birth certificates and marriage certificates establish the relationship; admission records establish the current classification.
Sequence a child's school or work plan before relying on a parent’s E-2 status, because a derivative child who ages out needs an independent solution. Hypothetical example: A Canadian bicycle-repair business purchaser expects her twenty-year-old daughter to manage the U.S. shop after graduation.
The daughter may study as a dependant, but the family must plan a separate work-authorized status before assigning her a paid management role.