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She does her shopping at the neighborhood grocery store.
Literal
She [topic-は] neighborhood [genitive-の] food-goods-store [at-で] shopping [object-を] does.
食料品店 is a transparent compound: 食料 ('food provisions') + 品 ('goods') + 店 ('shop'). The same productive compounding gives 文房具店 (stationery store), 靴店 (shoe store), 本店 (main store / bookstore depending on context). 買い物 is itself a noun built from 買う ('to buy') + 物 ('thing') = 'thing-to-buy / shopping.' The compound 買い物をする with light-verb する is the standard frame for 'doing shopping,' parallel to 勉強をする ('do studying / study') and 仕事をする ('do work / work').