I have this same problem, only now it is worse. Not only is there no car, but I placed the body down on the ground (to carry on doing other stuff in the hope it would be fixed later) and when I loaded back into the game the next day the body had dropped through the floor, so now I cannot pick up the body either.
I'm glad it is just a side mission and hope I am not missing something worth seeing/hearing in the story.
So far this has been my only quest breaking bug. Hopefully it stays that way and they eventually fix bugs like this. Though I suspect this quest will remain forever unfinishable for this character, as I have done far too much to go back to a previous save now.