Can I use Bill of Materials to order components online?
Most vendors have import BOM functions, and as long as you have a Manufacturer's part number in there, and the quantity, the import process is pretty much painless.
That is why it pays to spend some time creating a good part library.

I have used Digikey a few times, and it works quite well. If I remember correctly I also used Farnell/Newark, and those worked fine as well.