Nope, a reimbursement is not taxable.
Your biggest vulnerability is that IRS would deem your shopping activities to be a "hobby" rather than "self employment". As "self employment" you can deduct business mileage and other business expenses. As "hobby" you get to deduct virtually nothing. This is why it is so very important to conduct your business in a "business-like" manner.
The 'quick and dirty', of course, for legitimacy is the number of years your business is profitable. (IRS expects 3 of the last 5 years to be profitable.) But recordkeeping is your other line of defense--a diligent, systematic, ongoing accounting of revenues and expenses. I suspect you don't need to be an IRS agent to recognize records that were created the night before taxes were due or the week before an audit.
Obviously if you do reimbursement only jobs, such as most restaurants, these will always be a loss due to the mileage deduction. You will need to perform enough fee jobs as well to overall show a profit for your business. This makes it more and more important to find jobs that pay a decent fee, not just a small fee and a big reimbursement. I do grocery shops that pay $7 and reimburse $10. While the jobs are paying jobs, with mileage reimbursement at 50 cents, there are only 2 I could do as orphan shops and 'make money' on them because there are only two within 7 miles of my house. This is where routes come in handy.
As an example, I did 4 grocery stores each on Monday, Tuesday and Wednesday. For those 3 routes of shops I drove a total of 127.5 miles, for a mileage deduction of $63.75, so my IRS "Profit" for the run is $84 in fees - $63.75 mileage deduction = $20.25 of taxable income for $120 in reimbursed groceries. If I didn't run them as routes, all but 2 would have been done at a tax loss.