If you're working out of your home, your city may require you to have a business license. The thing to remember is that if you are required to have a license, and DON'T get one, and you file a schedule C on your income taxes (as opposed to putting your shopping income on there as hobby income), your state might share that information with your town.
I'd do anything your locality requires you to do. You'll sleep better knowing you are "legal" and the penalties for not doing something are usually at least three times the cost of doing them right to begin with.
But if you want to consider your mystery shopping as a hobby, not a business, then you wouldn't need a business license (usually -- check with your town). Your income is still taxable, but your expenses are only deductible on your Schedule A, and only up to the amount of the income (you can't deduct a loss for a hobby).
Talk to your tax preparer about the difference, and if you get a blank look when you ask, "What is the difference between claiming this as a business or as a hobby?" then go find a tax preparer who knows what they're doing.
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I pray it does not occur that the last thing I did before I died was vacuum the house or eat broccoli.