L-2 status gives him employment authorization incident to status, so immigration permits any job. Teaching in a public school also requires state certification, which is a separate process with its own timeline and may not recognize a Saskatchewan certificate without additional steps.
Authorization from immigration, certification from the state
An L-2 spouse may work without applying for a separate document; the admission record showing L-2 spouse status is evidence of employment authorization, and a card can be requested for convenience. That covers the immigration side completely: he may teach, tutor, or take any other job. The barrier is professional.
Public-school teaching in every state requires a state teaching certificate or licence, and out-of-country credentials are assessed through evaluation, sometimes with examinations or additional coursework. Private schools set their own rules. He should contact the state education department where the affiliate is located as soon as the destination is fixed, because certification can take months and cannot start until he has certain documents.
Children in L-2 may attend school and may not work, and they lose L-2 at 21. If his teaching plans are central to the family's decision, confirm the certification path before she accepts the transfer.