"Yes, it is permissible to do so. What is forbidden is that a woman be in a closed area with a man where they will be in a private situation; this is the forbidden khalwah (the state of being alone with a non-related person) in Islam. However, sitting in the car alone does not constitute any forbidden khalwah (the state of being alone with a non-related person). Khalwah implies a context in which an unrelated man with a woman are alone with each other out of sight of others.
Also, as you said it is just locally then I assume this would not take long, and therefore you need no mahram (adult male relative) to accompany you in this short trip. However, if you want to exercise ihtiyat (caution) by having a mahram with you, there is nothing wrong with this, but if you didn't do so, you will not be doing something haram (impermissible.)"
[Note: For contrasting opinions, search the keyword: travel, or perform a general search for: driving.]