He is employment-authorized incident to L-2 status and may work once admitted with the spousal notation, but practising veterinary medicine requires a state licence, which his immigration status does not supply. Alternatively, Veterinarian is a listed TN profession and he could seek his own status.
Work permission and professional licensing are different questions
As the spouse of an L-1A transferee, the husband is admitted in L-2 status and under current policy is authorized to work incident to that status, without a separate employment authorization document, provided his admission record identifies him as an L-1 spouse. That permission lets him accept any lawful employment, including within the family's subsidiary. It does not itself authorize veterinary practice: state veterinary boards set their own licensing, examination and credential requirements, which he must confirm with the board in the state where he intends to work.
He should begin that process before relocating. He also has an independent option: Veterinarian appears on the USMCA list with a qualification of a degree or a state or provincial licence, so as a Canadian citizen with a U.S. clinic offer he could obtain his own TN, which would survive if her L-1A ended.
Their children in L-2 status may study but may not work, and each child's twenty-first birthday ends derivative status.