I also favor each shopper keeping their own checklist. This is a business and it's good business to keep a ledger of your customers. I am assuming each of us keeps a ledger of payments due when we work for a company, so it's a simple matter to capture all companies, even those we haven't yet worked for.
Right now there's a feature here on the forum that tracks companies we are registered with if we activate it but I don't know how it works. When I go to the list of Mystery Shopping Companies and I click the name of a MSC, it brings up The MSC, the website address, and discussions about the MSC. For example, if I click Market Force, directly under the name and website address there is a question: Are you registered with Market Force? Yes, I am. No, I am not. If I click Yes, I am, is there some tracking of my answers?
I keep my own list and I would not use a list here on the forum so it's irrelevant to me, but this feature might be what the OP is hoping for.