I have never heard of Target INC as a mystery shopping company. My sense is that this is a scam using the Target name. To the best of my knowledge, when Target has mystery shops of their own facilities, they are performed by legitimate mystery shopping companies that most mystery shoppers are signed up with. I would provide these guys with no information whatsoever.
1) Mystery shopping does not send you funds up front to do shops--usually you are paid and reimbursed for your required purchases 2-12 weeks after the shop.
2) 99.9% of legitimate mystery shopping companies have a website for you to go and register to be a shopper with them.
3) If the deal is 'too good to be true' it is just that--not true. The vast majority of mystery shops provide between $5 and $15 in value (fee + reimbursement) to the shopper.
4) If you found them responding to a CareerBuilder or similar employment posting board or responding to an unsolicited email or to an ad on Craig's List, don't trust it.