A spouse admitted in L-2 status is employment-authorized incident to that status, which addresses the immigration side of working. Establishing and operating a business raises separate licensing, registration, tax and regulatory questions that status does not answer.
Authorization is one permission among several
Hypothetical example: the president's spouse runs a small livestock-hauling business from Picture Butte and expects to continue on the American side. The employment authorization that comes with L-2 status is real and useful. What it does not provide is operating authority for a carrier, state registrations, insurance, tax accounts or any sector-specific licence, and in a regulated industry those requirements can take longer than the immigration process.
Take them to the responsible agencies and to an accountant before assuming continuity. Children admitted as derivatives may attend school but are not employment-authorized on that basis, and derivative eligibility ends at twenty-one, which matters against a seven-year maximum. Confirm each family member's citizenship separately rather than assuming it follows the principal's.