Doubting Thomas Cook’s opening putrid stores (7)
I believe the answer is:
sceptic
'doubting thomas' is the definition.
(synonyms)
'cook's opening putrid stores' is the wordplay.
'opening' indicates taking the first letters.
'putrid' becomes 'septic' (similar in meaning).
'stores' is an insertion indicator.
The initial letter of 'cooks' is 'c'.
'c' put into 'septic' is 'SCEPTIC'.
(Other definitions for sceptic that I've seen before include "Person who doubts the truth" , "Person who tends to disbelieve" , "One questioning accepted opinions" , "Doubter; atheist" , "A doubting Thomas" .)