OP, with respect, you will have to learn to trust other people and step back a bit. I know it's horribly hard - my profoundly autistic son is now 15 and very vulnerable because of his LDs and autism. He'd go off with anyone, probably! He's also a flight risk and has no sense of danger. But in your situation, what are your choices? Trust the SEN transport in place for him, or take him yourself and essentially kiss goodbye to your day?
The LA providing the transport WILL provide an escort (to stop kids from fiddling with the doors, make sure they're calm, prevent them from overstepping the mark with one another, etc). Escorts are meant to be DBS checked and have done training. (I think this is an online thing, but with any luck they'll have done this job before so will know a bit about SEN.)
You'll probably be asked by the LA to provide some information about your son, his likes, his challenges, and any obvious risks. Plus your contact details. In our LA we have a transport 'passport' that clips onto a school bag. Taxi drivers/escorts are meant to consult this. In my experience, they don't always do that, but the info is there for emergencies.
Get to know your driver and escort. Make sure you have their mobile numbers and give them yours. Don't rely on the taxi HQ to pass on information! Hopefully you'll have some consistency and it'll be the same crew, but if it's a different one, make sure they've seen your son's 'passport', they know his risks and tendencies, and that the doors are child-locked! And that you have each other's numbers.
Maybe give your son something to do in the car? Does he use a screen? Have a favourite fiddle toy?
I won't lie - SEN transport is our main cause of stress during the school year - when we hear about it over summer (usually a couple of days before term starts), whether we'll have the same crew, if they're turning up, and when ... it can take a few days for a new crew to settle in. Communication is key, so do make sure you all have each other's numbers, and try and maintain good relations with them. If I like our escort and driver and think they're doing a good job, I tend to get them little treats like Costa cards at the end of the school year/Christmas. Some of them, to be frank, have been absolutely rubbish. In which case, you get on the phone to the LA and explain why they need to find a different provider.
It does get easier. Most SSs are brilliant and full of expert teachers and TAs who have seen it all before and are incredibly passionate and committed to getting the best out of, and for, our children.