Swear, finding head of pigeon on shelf (6)
I believe the answer is:
pledge
'swear' is the definition.
(I know that swear is a more specific form of the action vow)
'head of pigeon on shelf' is the wordplay.
'head of' suggests taking the first letters.
'on' means one lot of letters go next to another.
'shelf' becomes 'ledge' (synonyms).
The first letter of 'pigeon' is 'p'.
'p'+'ledge'='PLEDGE'
'finding' is the link.
(Other definitions for pledge that I've seen before include "Promise, oath" , "Promise solemnly and formally" , "Deposit" , "Vow; security" , "Give one's word" .)