I live in Virginia, and I own an LLC. I am looking to purchase a car to use for business travel. Do I need to title the car in my LLC name or can I use my own name?

You take deductions for business use of a vehicle regardless of whether the car is titled to you or your business. The important part of taking deductions for vehicles is keeping the mileage records (regardless of whether you choose to deduct the mileage rate or actual expenses).