Do I need to pay self employment tax when doing a part time job delivering or picking up cars for a car dealer? I am paid by the mile for picking up vehicles.

Then you are self-employed and must pay the self-employment tax on any income more then $400.

Confirming the other answer, what you are doing constitutes self employment. 

Although it may be news to you that you “have a business” (and most of the other people who get their first 1099); The IRS considers anything on a form 1099-NEC (1099-MISC in previous years) to be self employment income. Enter the 1099 income, at Other Common income / income from a 1099-NEC . TurboTax (TT) will complete Schedule C  for you and allow you to deduct any expenses associated with this income. You'll also have to pay self employment tax (social security & Medicare) on any profits greater than $432. These types of payments are frequently done this way. Your "employer" is not the only one.

The IRS also considers undocumented cash income (no W-2 or 1099), for work performed, to be self employment income.