In this answer I see explained the fact that Americans (and other English speakers who have accepted some American usage) use the noun "store" in many situations where other English speakers would use "shop".
What etymological or historical reasons might explain why this usage does not seem to have extended to the verb? As the answer above notes, I can go to a shop or a store, but at either location I will be shopping, not storing. Why?
As noted, shop, as a verb evolved around the late 17th century when "to store" was already a well-established verb with a different connotation.
Store meaning "place where goods are kept for sale" is first recorded 1721 in American English (British English prefers shop).
- mid-13c., "to supply or stock," from Old French estorer "erect, construct, build; restore, repair; furnish, equip, provision," from Latin instaurare "to set up, establish;The meaning "to keep in store for future use" (1550s) probably is a back-formation from store (n.).
- 1680s, "to bring something to a shop, to expose for sale," from shop (n.). The meaning "to visit shops for the purpose of examining or purchasing goods" is first attested 1764.
(Etymonline)
I think the difference stems from the time the goods for sale were kept in the store or shop. A store can keep goods for a long time - like a village store with everything from potatoes to horse feed. A shop (I had to look it up) stems from old german and means a lean to or shed. Here a customer could buy goods that were only kept there for a short time (day) - i.e. like a shutterable market stall. To go to the shop for fresh produce and go to the store for non-perishables.
