my flight leaves at 6:30 am on Sunday, I'll arrive at your location around 4 am. Do the shuttles run that early?

I dont know if BWI actually watches this section. I also dont know for sure, but if the lot is open I imagine the shuttles do run.

Definitely

Sure do

Yes, the shuttles run 24/7. They run more frequently starting at 5 am, but you should not have any problem making your flight.

Yes.The shuttles do run at that time.