One way to find out is checking with JobSlinger. It's a free site, and you can define your location to see types of shops available (restaurant, grocery, tanning, etc.). Most will link you directly to the MSP that offers the shops.
Grocery chains tend not to be nationwide. Larger chains tend to go with larger MSPs and most of the larger MSPs have some chain or other somewhere in the country. Places like Trendsource and Market Force I understand have groceries in some parts of the country, Sinclair I think still has the survivor locations of a chain that has closed out in my market. Not all groceries are shopped and some stronger players shop not only themselves but their competitors from time to time.
Of course not, but it tends to be a requirement for shops to appear on Jobslinger. Remember that Jobslinger pulls automatically from SASSIE companies that choose to participate as quickly as the shops post on SASSIE. Otherwise Jobslinger pulls from job boards at Volition and MSPA where job postings require a specific and separate post from schedulers/companies. There are some companies that never have posted to job boards other than their own that I have seen.