Yes. An L-2 spouse is employment-authorized incident to status and may run his own business, subject to ordinary U.S. business registration and tax rules that are separate from immigration.
Spouse work authorization covers self-employment too
Spouses of L-1 principals are admitted in L-2 status and are considered employment-authorized without a separate application once their admission record carries the spouse annotation. That authorization is not limited to employment by others; a self-employed consultant, tradesperson or designer may operate in the United States. What follows from that are ordinary commercial obligations, such as registering a business, obtaining licences that a state may require for his trade and dealing with tax, none of which are immigration questions but all of which should be planned.
L-2 children are permitted to study, but they have no employment authorization. The family’s statuses run with the principal’s L-1A validity, so if her transfer is cut short or the seven-year maximum approaches, his business will need a plan of its own. Hypothetical example: a museum operations director transfers with a spouse who plans to open a small design practice.
The spouse’s L-2 work authorization depends on admission in the correct spouse classification, so the family should inspect the admission record before clients are accepted. The useful record is the unexpired admission document with the spouse notation, not the principal’s approval notice alone. Children’s school plans should also account for the end date on that same admission.