He can, and without a separate permission. The spouse of a treaty investor is employment-authorized incident to status and may work for any employer, not only the family business. Unmarried children under twenty-one accompany as derivatives and may attend school, though they cannot work.
One spouse's freedom, one long-term limitation
For a household moving on a single business, the spouse's ability to earn elsewhere is often what makes the first two years survivable, and here it comes with the status rather than requiring a separate application. Two matters need looking at further out. Children stop qualifying as derivatives at twenty-one, and a treaty investment can run for many years, so a family with a teenager should assume that a separate status will eventually be needed and plan the timing rather than react to it.
More significantly, this category has no built-in route to permanent residence; it can be extended while the business continues to qualify, but it does not turn into a green card. A family whose real aim is to settle permanently should be examining an immigrant route as a parallel project, with its own advice, rather than assuming the business will eventually deliver one.