Reimbursements are not taxable because to the tax code they are not income. Generally mystery shoppers do not file quarterly but rather a schedule C attached to their return and self employment taxes are calculated at that time.
Generally quarterly tax filings are required if your unpaid tax liability would otherwise result in underpayment penalties when you file your return. You are not subject to underpayment penalties if there is less than $1000 to be paid with your return OR you have paid in 90% of what you owe for the current year OR you have paid in 100% of what the previous year's total tax liability was.
If you are currently employed other than as a mystery shopper and usually owe a little bit in taxes you may want to increase your withholding a bit. If you are only working as a mystery shopper, between legitimate deductions for your business and then with the overall tax return having personal exemptions and standard deduction you are unlikely to pay much, if any, income taxes on mystery shopping income. If your mystery shopping has a net income of over $400 it will be subject to self-employment tax and the rate for that is about 15%. This is calculated and collected with your tax return and there are no penalties for not having paid into that system throughout the year.