Passport endorsement exists in Virginia (4)
I believe the answer is:
visa
'passport endorsement' is the definition.
(I've seen this before)
'exists in virginia' is the wordplay.
'exists' becomes 'is' ('be' can be a synonym of 'exist').
'in' means one lot of letters goes inside another.
'virginia' becomes 'va' (US state abbreviation).
'is' going within 'va' is 'VISA'.
(Other definitions for visa that I've seen before include "Travel pass" , "Travel document is required to enter Virginia" , "Travel warrant" , "Addition to passport" , "Mark on passport" .)