Starting in tax year 2018, you can no longer deduct employee expenses, including mileage and/or automobile expenses. Even in previous years, the deduction was limited to being an itemized deduction, subject to the 2% of AGI threshold.
If you are a self employed person, you can write off the business percentage of all you automobile costs (gas, and registration fees, depreciation, interest, maintenance etc) or just use the governments mileage allowance (58 cents). Most people use the mileage rate. You are expected to keep a log of your business trips, not estimates.
So, receipts are not 100% mandatory when you file. But receipts or a log are mandatory if you are audited.