I have the same problem. It's really annoying that they decided to stop using bags again. Tesco says they do bags or crate liners, and the last couple of times the entire crate has had one large plastic liner without handles. When filled with big bottles of milk, washing powder and tins, it was so heavy I couldn't lift it, and if I had tried I bet the liner would have split. I had to keep the driver waiting while I went off to find bags to put the heavier items in before I could lift the liner out of the crate.
Morrison's, ASDA and Iceland still do bags, and I find myself using them more often, after having had regular deliveries from Sainsbury's for years.
It's true that you can ask the Sainsbury's driver to move the items from the crate and into boxes for you (I use laundry baskets with handles) but it feels wrong to make their job more difficult.
I've asked the drivers if they are allowed extra time for deliveries now that crates have to be unloaded one item at a time on the doorstep. Since it was a much quicker process when customers only had to lift out the shopping in bags. They assure me that their schedules have not been adjusted to compensate and that it is very stressful for them.
I too have invisible disabilities and feel uncomfortable asking them to do the unpacking for me, but sometimes I am just too feeble to attempt it. The situation becomes even worse when the driver is one of those blokes who doesn't seem to understand that putting tomatoes and kiwi fruit in the bottom of the basket and then laying 2L bottles of milk and 4 packs of baked beans on top of them is not advisable. The last time I had to keep saying "No, please, put the tins and heavy items in the bottom of the basket, they won't be crushed when you put the eggs and biscuits on top." I can't decide if they are just unimaginative, or if it is a passive aggressive protest, like the husband who cunningly boil washes your delicate clothes to get out of being expected to do his share of the laundry.
I wish they'd employ more female drivers. Iceland seems to manage to find women delivery drivers, so why don't the other supermarkets? It's so much less stressful dealing with women.