Not on the basis of TD status. A dependant may accompany the principal and may study, but is not employment-authorized, and unpaid work is not automatically outside that restriction. The activity itself has to be examined.
Unpaid does not mean unregulated
Hypothetical example: the veterinarian's spouse is a registered veterinary technologist and offers to help at the practice without pay while looking for something permanent. Two separate problems arise. The first is that describing work as volunteering does not by itself place it outside employment restrictions, particularly where the work is of a kind the business would otherwise pay someone to do.
The second is that veterinary technology is itself regulated, so performing it without the destination state's credential raises a licensing question independent of immigration. If the spouse wants to work, the honest routes are their own status permitting it, or waiting. Children admitted as dependants may attend school; derivative eligibility ends at twenty-one, and each family member's citizenship should be confirmed individually.