No, in order to use the standard mileage deduction, you must own the vehicle.
You will need to use the actual expense method in order to claim vehicle expenses. You will still need to keep track of the miles driven for business purpose and the miles driven for personal purposes. The percentage of business use will then be calculated from the mileage numbers. You would be able to claim the business percentage of the rental fee as a business expense, along with other actual vehicle expenses such as gasoline.
See the following information from the IRS to learn more:
Topic no. 510, Business use of car