No, I'm pretty sure this is a scam. Legitimate mystery shopping companies do not contact strangers and offer to pay them to do mystery shops. To work with a legitimate mystery shopping company, you must sign up on their website and provide your SS# or EIN# and your personal information, because any work you perform is taxable. Legitimate companies, with few exceptions, do not pay in advance. You do the work and you wait to get paid back. And pay for shops generally ranges from $5-$30.
Let me guess - they send you money. You cash the money gram or deposit it in your bank account. You send some of the money to another person through Western Union. Then you do a mystery shop and keep the rest of the money. And when you find out the money gram is no good, you will owe the bank the whole amount.