Just try a few different supermarkets, one will have a suitable slot, even without priority access.
Or do as advised above and register with the government as a shielder.
I think you get a guaranteed weekly slot with supermarkets like Sainsburys etc then.
In the delivery instructions, write that you are shielding and need a contactless delivery.
Then they either just leave the carrier bags on the doorstep or if you don't want carrier bags leave a couple of boxes on the doorstep for the delivery drive to unload the groceries into.
When they ring the bell, just wave through a window since you have selected contactless, shouldn't even be necessary to open the front door until they are back in van/gone.
Also if they email you to say there are substitutions, if you don't want them leave a note on the door or in the boxes specifying which ones so you don't have to go face-to-face to deal with that as well.
The drivers are supposed to wear masks I think, but certainly in reality many don't. So feel free to ask them to or not open the door etc.
If you are going carrier-bag-free and leaving boxes outside, put them out up to 1 hour early, as often they come early without calling first and it can be stressful rushing around sorting getting boxes onto the step while they are stood there waiting without a mask on