Eastern chaps going west, initially sealing one’s fate (7)
I believe the answer is:
nemesis
'fate' is the definition.
(nemesis can mean someone's doom or fate)
'eastern chaps going west initially sealing one's' is the wordplay.
'eastern' becomes 'e' ().
'chaps' becomes 'men' (chap can mean a man).
'going west' says the letters should be written backwards (letters heading to the west or left).
'initially' suggests taking the first letters.
'one's' becomes 'i's' (Roman numeral).
The initial letter of 'sealing' is 's'.
'e'+'men'='emen'
'emen' reversed gives 'neme'.
'neme'+'s'+'is'='NEMESIS'
(Other definitions for nemesis that I've seen before include "Divinity" , "Unbeatable rival" , "Come-uppance" , "Vengeful justice" , "Greek god of vengeance" .)