The closest thing to an "accredited" company is the MSPA list. The MSPA is a trade organization of mystery shopping companies, there called providers. A company, or provider, pays a fee to become accredited there. I also believe there is a requirement of five years in the business as well. For shoppers, this list does give some measure of comfort because the companies have been around and choose to associate with the major trade organization.
With that said, there are some fine companies that are not MSPA members. The "Official List" linked at the bottom of every forum page lists lots of them. Consider the larger ones and also see what other shoppers say about them. A great reputation with shoppers is perhaps the best validation of any mystery shopping company.
Happily shopping Rhode Island and nearby Massachusetts and Connecticut