This is a great question. So, the answer is Maybe.
DOL does not specifically address this in statute. A state agency may, however. Many employers have been advised to terminate the employer sponsored program and contract with a company like Uber or Lift to offer this benefit. The 3rd subheading of this article makes an interesting statement.
https://www.cbia.com/resources/hr-safety/issues-laws/wage-hour-issues-travel-time/
In addition to the pay concerns, there are other employer liabilities that may tie back in and cause a wage and hour question to be raised.
Here is a reference that makes some great points citing several court cases. https://www.business.com/articles/employee-carpooling-incentive-legal-liability/
To be safe, I would say pay at least minimum wage for time spent driving plus gas or mileage. Count this as hours worked for overtime purposes, as well.
June 2018
Travel Pay