An L-2 spouse is employment-authorized incident to status and may work for any employer without a separate application for permission to work. Children hold L-2 status as dependants, may attend school, and remain dependants only while unmarried and under twenty-one. Professional licensing is a separate matter from immigration status.
Permission to work is not permission to practise
The immigration half of this question is generous and the professional half often is not. Employment authorisation incident to status means the spouse's right to work follows from the status itself. What that right does not do is satisfy a state licensing board, and families relocating around a spouse's regulated profession sometimes discover the sequencing problem only after arrival.
Hypothetical example: the spouse of a transferee from Sainte-Cécile-de-Milton is a pharmacist who has practised in Québec for nine years. She may take employment in the United States as soon as she holds L-2 status, but she may not dispense until the board in the destination state has licensed her, which will involve examinations, a review of her programme and often supervised hours. The useful step is to write to that board before the family fixes a moving date, because the answer determines whether she arrives able to work in her field or in something else.