Old boy's meeting priest on street, becoming most vague (9)

'most vague' is the definition.
(obscure can mean vague or unclear)

'old boy's meeting priest on street' is the wordplay.
'old boy's' becomes 'OBs' (abbreviation for a private school 'old boy').
'meeting' says to put letters next to each other.
'priest' becomes 'cure' ('priest' in French).
'on' says to put letters next to each other.
'street' becomes 'st' (abbreviation).
'obs'+'cure'+'st'='OBSCUREST'

'becoming' is the link.