TD status carries no U.S. employment authorization, and working while physically present in the United States raises questions even where the client and the payer are Canadian. This is a fact-specific area and it should be reviewed before the move, not assumed.
Where the work is done matters
Hypothetical example: a Hampton spouse keeps the books for several local farms and small contractors and plans to keep doing so from a laptop after the family relocates on the principal's TN. The instinct that this is purely a Canadian arrangement is understandable and is not the whole analysis, because activity performed while physically present in the United States is what the question turns on, not the location of the client or the bank account. TD status confers no employment authorization of any kind.
Some remote arrangements for a foreign employer are treated differently from others, and the distinctions are genuinely technical, which is exactly why this belongs in a professional review before the family commits. The unsafe path is to assume the answer, build household income around it, and discover the position at a border crossing.