If he accompanies you in L-2 status he is employment-authorized incident to status and may work for a US employer. Continuing to perform his existing Canadian job from inside the United States is work performed on US soil and should be assessed on its facts rather than assumed to be outside the rules because the employer and the pay are Canadian.
Decide the second job before the move, not after
Households commonly assume that a remote job for a Canadian employer travels with them untouched. The safer approach is to identify the status the spouse will hold, confirm what it authorizes, and then have the arrangement reviewed for the particular facts: who the employer is, where the work is performed, and how it is paid and taxed. An L-2 spouse who is employment-authorized has the widest room, which is one practical reason a household may prefer an L petition over a category whose dependants cannot work.
Tax residency, payroll withholding and social security are separate questions again, and they should be put to an accountant familiar with cross-border employment before the first pay period rather than at the end of the year. Hypothetical example: a museum conservator's spouse intends to accept freelance illustration work after the move. Before accepting any contract, the household should confirm the status held and preserve the records an employer or client will reasonably request.
Make a separate list for work authorization, school registration, professional licensing, taxes, and dependent-age dates. The first review should produce a lawful plan for each family member. The error is treating a remote or freelance arrangement as immigration-neutral when the work is physically performed in the United States.