One grabbing chest ran off (8)
I believe the answer is:
snatcher
'one grabbing' is the definition.
(someone who snatches or grabs something)
'chest ran off' is the wordplay.
'off' indicates an anagram.
'chest'+'ran'='chestran'
'chestran' is an anagram of 'SNATCHER'.
(Other definitions for snatcher that I've seen before include "Person grabbing" , "Thief" , "See 1" , "One seizing suddenly" , "Stealer; kidnapper" .)