Travel between work locations for the same enterprise is deductible.
If you have a home office for your mystery shop business (and it doesn't have to be a room; see IRS website for definition of a home office) and you leave that home office to go do a shop, you are traveling between two work locations.
If you have a home office for your mystery shop business and also a W-2 job, travel from the home office to your W-2 job is not for the same enterprise and is not deductible. If you then go from work to the McDonald's down the road at lunch time to do a shop, that is still travel between two different enterprises and not deductible.
But if you went from your home office to the McDonald's for a breakfast shop, those miles are deductible.
And then you can go down the road to your day job (non-deductible) and then to a hardware shop after work (non-deductible) and then home again (deductible) to your home office.
Yes, the rules are a bit stupid on the surface. But if you understand how it works, you can work it to your advantage. But the key is to have that home office for mystery shopping. And if you don't have a day job to interfere with this, you can leave home for a route of mystery shops, stop at the grocery store on the way home and the whole trip is a tax deduction. Minor personal errands can be run on a business drive as long as you don't divert more than a half mile or so out of your way.
And do go read about this on the irs website. This is just a quick summary of some of the nuances.
And I have two home offices; one is rented to my corporation for my tax business, the other is my other desk and other computer and is used for mystery shopping. I'm driving between work locations no matter what i'm doing.
Time to build a bigger bridge.