I mean, I've definitely had flu jabs at pharmacies in supermarkets, so why not?
It's become a key part of the strategy in the US. As I mentioned up-thread my daughter is getting hers at the US affiliate of Boots, and my son had his at one of their other locations last weekend. They can't do the numbers of jabs that the mega state-run sites do, but there are now thousands of local pharmacies doing them, and it's very convenient. Most of the big supermarket chains here have pharmacies in the store, so supermarkets are doing them here too.
Some years ago the regulations were changed to allow pharmacies to do vaccinations more easily, mostly to increase the uptake of the annual flu vaccines. So the big pharmacy chains here invested heavily in cold storage, staff training, software improvements, etc., and were totally ready to jump in and do covid vaccination. It's really ramped up in the last month or so with the greatly increased supply from Pfizer and Moderna.