Yes, we submit proposals whenever the PI DEMANDS it.  2 am,  sure that's fine.

Our manual indicates a 2 week lead time to get the proposal materials to our of
fice.  By proposal materials, I mean the narrative and biographical sketchs and
 appendices.  Our office does all the rest.  We do the budget and forms, face p
ages, copying, mailing, tracking etc.  It is extremely rare for a faculty type
to bring a completed package ready to go several days prior to a deadline.