If you're really interested in becoming a mystery shopper, I would recommend you check out the Mystery Shopping Providers Association. They have a certification program for mystery shopper wannabees, and it appeared that many companies that I looked it did inquire if applicants had completed this course. I would check with the agency in advance on their policies.
Mystery shopping is a legitimate business that is primiarily done through agencies. I am not a mystery shopper and so I'm unable to suggest how to go about it. I would recommend posing any question you have at the MSPA forum.
From what I have learned, this job is not nearly as glamorous as it sounds and the pay isn barely minimum wage--according to MSPA president Mike Green the average pay for a fast-food restaurant or a bank audit is about $10 to $12. It also requires good writing skills, computer knowledge, and strong observation skills. Remember, this is a job and treat it like you would any other employment. Be profressional in your inquiries and check the company out thoroughly.
The following links connect you to agencies that are either members of the American Marketing Association (AMA) or the Mystery Shopping Providers Association (MSPA). I am not affiliated with any of these companies and do not consider their placement on these pages as a recommendation from me.