No, in most cases you cannot ride with the carrier. This is why:

With local towing, it may be pretty common to ride along with your vehicle as it heads back to the repair shop or dealer when you call someone like AAA. Local towing usually is 1 car with a conventional drag-n-hook tow truck. They are taking you right from where they got you to where you are going.

However, auto transport is multiple cars. As a result, it's extremely rare. In very short distance transport we've seen the driver of the carrier do the favor. But that is never guaranteed and should not be counted on being your situation. And we definitely do not see it for long distance. This is because the carrier doesn't just go from point a to point b.

They are usually picking up cars at one location, going to another to get another vehicle, and then another. Same thing for drop off. Let's say it was possible, you would lose an entire day or even more, as well as likely be pretty unhappy about that. Also, it's not a stretch to assume the carrier might not be too keen on having a stranger sit with them in the passenger seat for all those pickups and drops, let alone the days they will be on the road.