If your company does not reimburse you for miles driven, then nobody else will, including the IRS. But you can claim business use of your vehicle and claim the "per mile" deduction. Note that you will need to keep detailed records of *all* miles driven on your car - not just business miles.
Since this is for your W-2 job (I assume) you'll claim your vehicle business use under the deductions & credits tab in the Job Related Expenses section.