Yes it is self-employment. You cannot claim mileage, per se, but you can claim a portion of fuel receipts, car insurance and repairs. You do have to be careful, in that you cannot count the mileage driving to the job, as it is considered driving to work, which is not deductible, but you can count the mileage driving from one retail outlet to another. You can kind of get around this rule if you have enough income to absorb a home office expense deduction.
All you really need is a calendar on which to write the mileage driven on each day you worked as a mystery shopper. You need the starting mileage on the car for the year and the ending mileage in order to calculate the actual percentage used for business. That determines how much of your vehicle expenses are deductible.