Yes but its still a very male dominated country where many men still hold tight to religious law. Everything ive read tells women travelors and expats. to be very careful. Something as simple as a speeding ticket could land you in serious trouble. It might be safer to hire a male to make the purchase for you and a male driver. Most likely salesmen are going to be male which means you may not be taken seriously or you may get cheated.
I believe that women can legally own cars among other property in Bahrain. However, most of them often opt to authorize their male counterparts to manage it for them. Read more from http://www.refworld.org/docid/47387b6a23.html.