Not automatically through the purchase. A qualifying E spouse is generally employment-authorized incident to valid status, but that depends on the spouse holding E dependent status and being able to evidence it, for example through the I-94 annotation or other applicable proof. Children admitted as dependents gain no work authorization, whatever their role in a family business.
Check the spouse's status and proof before payroll starts
Payroll is where this becomes concrete. Before adding a spouse to the staff list, verify the dependent classification and the document the business will rely on for employment verification, and keep that check separate from the principal's case. If a teenage child is expected to help at the counter, treat that as unauthorized employment unless a separate route is examined.
Note each family member's nationality, current location, and intended arrival date alongside the closing schedule. Two points are easy to overlook once the payroll question is settled. Children admitted as dependants may generally study, but that permission does not extend to employment however informal the setting, and a weekend role in the family business is exactly where this gets tested.
Separately, a dependant's own nationality can change the procedure for obtaining their status even where the relationship itself is straightforward, so record each passport rather than one household nationality. Hypothetical example: a specialty cheese producer's two teenage children expect to help at farmers' market stalls, which the family treats as unpaid family activity only after taking advice, rather than assuming it is permitted.