It is the taxpayer's responsibility to prove any deduction. Most taxpayers are never audited. If you are unlucky enough to get audited, and you can't document your mileage deduction, the examiner can recalculate your expenses and tax owed based on whatever you can prove, and send you a bill for the extra tax.
Even so, I expect your delivery apps at least track the mileage you drove for each delivery. That will be a start, although they might not track miles driven while not officially on a job, even though those miles are usually deductible as well.
The rules for deducting vehicle expenses are in chapter 4 of publication 463.
https://www.irs.gov/pub/irs-pdf/p463.pdf
The standard mileage method is better for most drivers, and is less paperwork. You can't track your first trip of the day from home to your first work location, or your last trip home at the end of the day, since that is considered "commuting" and is not deductible. But other mileage is deductible even if you are not on the clock. For example, if you give someone a ride home from a bar late at night, and you drive back to the part of town where all the clubs are to be ready for another pick-up, that mileage would be deductible even though you don't have a client in the car.
