It is not possible to deduct these vehicle expenses without being classified as self-employed.
If the 1099-Misc is reporting the income in box 7 - Nonemployee compensation, it is considered "Income from Self-employment", and you'll need to report the income on Schedule C of your return. You can then deduct your vehicles expenses (gas/mileage) on that form.
If you received this 1099-Misc as an employee (income reported in box 3 - Other income), you will not be able to claim the gas/mileage expenses on your return due to changes in the tax law. If this is the case, you report the entire amount as "Other income" on your return.
