I'm finding it a little difficult to follow your argument. I dont see anyone arguing for reckless lorry driving in Forest Hill - just the opposite. The store has been in its current position for many years and has always been serviced from Pearcefield Avenue as well as from London Road so there have been delivery vehicles there for some time. I recall some resident action some years ago in respect of early morning deliveries and concerns about noise and vibration. I am in that area frequently, several times a week and have never experienced or witnessed any problems with delivery vehicles. I have however witnessed cars going the wrong way down Manor Mount at speed and police cars going the wrong way up Davids Road as well as general motorists. There are regular delivery trucks to the Hob who reverse up from the south circular- that is definitely illegal. If delivery vehicles to Sainsburys are committing traffic offences then this surely needs to be reported to the police as such with dates, times and registration numbers etc. The Council sometimes park their mobile cctv unit at the top of Manor Mount; you could try and persuade them to monitor the situation lower down- they are very receptive to resident pressure and the local councillors would probably also be supportive.
I still can't see how Sainsburys can be held liable for third party offences carried out by drivers other than direct employees - its just outside their jurisdiction- they can only at their own behest use reasonable endeavours to request compliance. Its also not clear whether all these trucks are servicing Sainsburys and not other high street businesses.
Ultimately we need this business to remain in London Road and Forest Hill as it is the lynchpin of the local economy which encourages most people to shop in this area in the first place.