As Toffler says, "I don't have a car" is correct and would be the one you would hear almost all the time.
Also correct, but far less common, is "I have no car." This is older English but would be understood. Notice that it is your second sentence without the "a" article.
Also conveying the same meaning is:
"I'll ride my bike", "I'll walk", "I'll be taking a taxi", and "You'll have to come get me."