is this a legitimate business expense?
No. Vehicle use of a non-business vehicle not owned by the business prior to being officially "open for business" is not a legitimate business or start-up expense.
Don't get confused with what you read in Publication 535 about travel costs being start-up costs. That refers to things like plane tickets, bus fares, cab fares, etc. and other items related to transportation where you can provide solid proof that "this trip cost $X dollars and was for the sole purpose of getting my business off the ground."