An L-2 spouse is employment-authorized in the United States incident to status, which covers working there. Continuing to operate a Canadian business while living in the United States raises tax and, depending on how the work is performed, other questions worth reviewing first.
Two countries, two sets of consequences
Hypothetical example: the transferring sibling's wife runs a small equine supply business from Hampton and expects to keep it going remotely. Her L-2 status authorizes employment in the United States incident to that status, so working there is not the obstacle. The complications sit elsewhere.
Canadian and U.S. tax residency, where the business is managed from, whether it creates a U.S. presence, and payroll or sales tax obligations in a new state are all real questions, and they are for an accountant qualified in both countries rather than for an immigration file.
The sensible sequence is to get that advice before the move, decide whether the business is being relocated, wound down or genuinely left in Canada under someone else's day-to-day management, and then describe the arrangement consistently everywhere.