Vicar has no head for crime (5)
I believe the answer is:
arson
'crime' is the definition.
(crime of illegally starting a fire)
'vicar has no head' is the wordplay.
'vicar' becomes 'parson' (both are members of the clergy).
'has no head' means to remove the first letter.
'parson' with its initial letter removed is 'ARSON'.
'for' acts as a link.
(Other definitions for arson that I've seen before include "Crime of fire setting" , "Roans (anag)" , "Criminal burning of property" , "Incendiary crime" , "Crime of fire-setting (5)" .)