Vehicle noise starts to annoy local church dignitary (8)
I believe the answer is:
cardinal
'church dignitary' is the definition.
(I've seen this before)
'vehicle noise starts to annoy local' is the wordplay.
'vehicle' becomes 'car' (car is a kind of vehicle).
'noise' becomes 'din' (din is a kind of noise**).
'starts to' suggests taking the first letters.
The first letters of 'annoy local' is 'al'.
'car'+'din'+'al'='CARDINAL'
(Other definitions for cardinal that I've seen before include "Vivid red of a senior bishop?" , "fundamental importance" , "Red American songbird" , "Most important number" , "Main - church dignitary" .)