Can my employer require me to use my personal vehicle during work hours when my job description has changed?

Unfortunately, employers are not required to reimburse employees for expenses incurred in connection with their work, including travel expenses. There are exceptions; there can be no: discrimination involved in an employer's decision not to reimburse an employee; union/employment contract which provides for such reimbursement; company policy which mandates reimbursement (or specific employer representation, promise, etc. as to repayment of such expenses). Other than these exceptions, reimbursement is at an employer’s discretion.

Note:  If you are not reimbursed, then your travel expenses may constitute expenses which you can deduct on your own taxes. You should consult with your tax preparer or accountant about this. Additionally, if you have to travel outside their normal shift and you are a non-exempt employee, you are entitled to overtime for any hours worked over 40. 

As for losing your job if you refuse, as an "at will" employee you can choose to work for an employer or not. And your employer can fire you, with/without notice, for this reason or any reason at all (subject to the 3 exceptions noted above).

Bottom line, either speak to your boss, HR, etc. and explain your side of things or else pay if you want to stay. While unfair it is the law.