As far as I am aware there are a number of cars that you can purchase in the singleplayer that are bulletproof. Have you checked the wiki or rockstar's own website about this?
The first car that comes to mind is the Karuma, its an average sized saloon car that is completely bullet and rocket proof including the tires.
Here is the link to the wiki that details all the "Armoured" cars available. You can peruse them as you need.
I'm not sure if you can buy all of the listed cars in the single player but I know that you can definitely buy most. Just go to the web browser on your phone and then onto the car supplier that you want.
You will need to have purchased a garage before hand otherwise the vehicle will not be delivered to you.
EDIT: You can also find a duke o death in a rusted out storage shed at the very southern tip of the Murrieta Oil Field in southern Los Santos, free of charge. After this you can buy the car from the Army website. Link to the forum post about this.