Shopper initially rushed to shop (5)
I believe the answer is:
store
'shop' is the definition.
(synonyms)
'shopper initially rushed' is the wordplay.
'initially' says to take the initial letters.
'rushed' becomes 'tore' (to tear around is to rush or run fast).
The initial letter of 'shopper' is 's'.
's'+'tore'='STORE'
'to' is the link.
(Other definitions for store that I've seen before include "Reserve, shop" , "Large shop; computer memory" , "Shop; repository" , "Depository for goods" , "Eg, granite" .)