The short answer is: no, you normally can’t ride with your movers in their truck. Most moving company policies prevent it because of liability concerns and insurance limitations.
Also, many moving trucks only have enough seats for the professional team, and if there’s no room, you won’t be able to ride in the cargo area either. Plenty of states have laws prohibiting passengers in cargo areas.
The Federal Motor Carrier Safety Administration (FMCSA) states that you can only ride in a cargo area if it's designed to protect passengers in the event of a crash. Because moving truck cargo areas typically don't have seats, seat belts, or other protections, it’s not legal—or safe—to ride in them.