No. TD dependants may live in the United States and may study, but TD status does not carry employment authorization for a spouse or a child.
TD study is permitted; TD work is not
Hypothetical example: Households planning around a single seasonal income sometimes assume a spouse will find work once settled, and the assumption is wrong for this category. A TD spouse may enrol in study, and children may attend school, but neither may accept employment on the strength of the dependant status alone. A spouse who intends to work needs their own basis for doing so, which means their own qualifying offer and their own application, assessed on their own facts.
That is a separate matter with its own timeline, and it will not be resolved by the principal's approval. Remote work performed for a Canadian employer while living in the United States is not a safe workaround either; it raises questions about the nature of the activity that should be assessed specifically rather than assumed. The practical step is to declare the intention early, so the household budget and the immigration plan are built on the same facts.