The setter's people start to exult following negative vote for candidate (7)
I believe the answer is:
nominee
'for candidate' is the definition.
(synonyms)
'the setter's people start to exult following negative' is the wordplay.
'the setter's people' becomes 'mine' (I can't explain this - if you can you should give a lot more credence to this answer).
'start to' suggests taking the first letters.
'following' says to put letters next to each other.
'negative' becomes 'no' (no is a kind of negative).
The initial letter of 'exult' is 'e'.
'mine'+'e'='minee'
'minee' put after 'no' is 'NOMINEE'.
'vote' acts as a link.
I am not very happy about this link. It may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for nominee that I've seen before include "One named as candidate" , "Person put up for a post" , "One put up for office" , "somebody put up" , "Candidate - appointed person" .)