Tries initially expediting some late vehicles (7)
I believe the answer is:
hearses
'some late vehicles' is the definition.
(vehicles which carry 'late' people)
'tries initially' is the wordplay.
'tries' becomes 'hears' ('hear' can be a synonym of 'try').
'initially' becomes 'es' (I can't explain this - if you can you should believe this answer much more).
'hears'+'es'='HEARSES'
'expediting' is the link.
I am not very happy about this link. It may be part of another bit of the clue.
(Other definitions for hearses that I've seen before include "Cars for corpses" , "Vehicles for carrying coffins" , "The last carriers" .)