It depends on the Eurotunnel service you are using.
Yes. The Eurotunnel Vans on Passenger (VP) service lets van drivers travel in the comfort of their own vehicle on the passenger trains. You can book to a specific time slot, and travel via the dedicated VP lanes missing the peak Freight traffic allowing you to meet those timed deliveries.
No. Eurotunnel Freight has an obligation to maintain the safety of lorry drivers. All truck drivers are required to travel in the Drivers' Club Car during a crossing. It is strictly forbidden for any drivers or passengers to remain in the cab.