When you receive money as an independent contractor they do NOT take out out taxes, FICA etc. Most of the companies will not even send you a 1099 at the end of the year unless you have earned more than the minimum amount (500? 600?) in fees from this. This does NOT mean, however, that they are not claiming the fees they paid you and the reimbursements they sent you as THEIR business expenses, so this stuff is not free money.
You will file your mystery shopping as a business using a Schedule C with your regular tax return. That form has you enter all business income and deduct all business expenses. There are two ways you can file--as a "cash" business or as an "accrual" business--and once you have done it one way the first year, you must do it always the same way. I choose "accrual" which keeps all the income and expenses in the same year and it means I claim as income all fees and reimbursements for a particular year even if I don't receive the actual payment until the following year. Many folks choose to file as a "cash" business where they have what they actually receive and what they actually pay in a particular year be their income and expenses.
I claim all fees and reimbursements as income and then deduct from this total all of my expenses--the reimbursements from shops, the unreimbursed expenses, the mileage, my office supplies and equipment costs, etc. As an "accrual" taxpayer I have some leeway to purchase office supplies and equipment before the end of the year if necessary to keep my net income from my business below $400. It is in my interest to have my business make SOME money every year (IRS only requires that you have a profit 2 or 3 out of 5 years to keep this from being categorized as a "hobby" which would significantly reduce the deductions you are entitled to). It is also in my interest to have my net income from this business be LESS than $400 so I do not have to pay social security and medicare taxes on it (as a self employed person, if you have to pay social security and medicare taxes you are paying both the employer and the employee portion of those taxes so it mounts up quickly).
If you are not filing jointly with a spouse it may make sense to send in some small quarterly tax payment to IRS using their estimated payment program, but chances are that you will find your legitimate expenses, if you keep careful track of them, make your net income insubstantial. For example, it probably does not feel like it costs you 50.5 cents per mile to drive your vehicle to do shops because even if you weren't shopping you would have a vehicle and be paying insurance and licensing for it. You are more likely to think of a trip for a shop that is 8 miles away (16 round trip) as "costing you" a gallon of gas--or $4 to do the shop. IRS, however, is allowing you to deduct $8.08 as a business expense for that trip because their number includes allowance for gas, the vehicle itself, repair, maintenance, insurance, licensing and any other expenses except tolls and parking (which they will allow you to deduct separately).