Sure, I'm white and come from a disadvantaged background and didn't have a problem. Granted, there were much more URM, but still a fair amount of white/ORM as well. FWIW, I got rejected from Columbia almost immediately so some schools may only want URM, can't speak for all of them.
I applied in mid-February. I would suggest applying as early as you can, but if you have no grades there's not much you can do. You can start your application and submit it before sending the transcript though, so that would probably be the best idea. The application's a bit long.