Yes. An E-2 spouse is employment-authorised incident to status and may work for any employer, or in the family business.
The spouse's employment freedom is one of E-2's practical advantages
Hypothetical example: A spouse admitted in E-2 dependent status holds employment authorisation by virtue of that status, without needing a separate grant of permission to work, though an employer may still want documentary evidence of it. The spouse may take outside employment or work in the invested business itself. Children in E-2 dependent status may study but are not employment-authorised, and they cease to qualify as dependants at twenty-one.
Two points are worth planning around. First, if the spouse will work in the business, that involvement should be described consistently with the control story told about the principal investor, because a file that gives operational direction to both without explanation reads as confused. Second, the dependants' status depends on the principal's, so a decision to sell the business affects the whole household's permission at once and should be taken with that in view.