No. The dependent status that accompanies a TN carries no work authorization at all. She may live in the United States and may study there, and the children may attend school, but any employment requires a status of her own that permits it.
Study is allowed, earning is not
Dependent status exists only for as long as Wendell holds his own, and it authorizes no employment of any kind. That includes an American job, freelance work, and continuing to work remotely for an Ontario employer from a house on the other side of the border, which households in seasonal industries often assume will be tolerated. Two further points shape planning.
The relationship must be a legally recognised marriage; a long-standing common-law partnership is not a spousal relationship for this purpose and needs its own route. A child stops qualifying as a derivative on turning twenty-one, so a family with a nineteen-year-old should be counting the months now rather than discovering the problem at a renewal. Where the second income is not optional, it is usually better to ask whether the spouse could hold a status in her own right than to hope the restriction will go unnoticed.