Rural Carriers are employees, not sub-contractors, therefore not Self-Employed.
Rural Carriers are paid a "Qualified Reimbursement" for the use of their vehicles and that amount is NOT included in Box 1 of their W-2, therefore it is not taxed.
Pre-tax income cannot be deducted.
If you are not Self-Employed, you cannot depreciate a vehicle.
According to the IRS:
“…If you were a rural mail carrier, you can treat the amount of qualified reimbursement you received as the amount of your allowable expense. Because the qualified reimbursement is treated as paid under an accountable plan, your employer shouldn't include the amount of reimbursement in your income. You were a rural mail carrier if you were an employee of the United States Postal Service (USPS) who performed services involving the collection and delivery of mail on a rural route. Qualified reimbursements. These are the amounts paid by the USPS as an equipment maintenance allowance under a collective bargaining agreement between the USPS and the National Rural Letter Carriers' Association, but only if such amounts don't exceed the amount that would have been paid under the 1991 collective bargaining agreement (adjusted for changes in the Consumer Price Index since 1991 as detailed in section 162(o)(3)). If you were a rural mail carrier, do not use Form 2106. Your employer should not include the amount of reimbursement in your income.”
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