She might have gone out to do some shopping.

Literal

She [topic-は] shopping [purposive-に] went-out [might-be-かもしれない].

買い物をしに combines the noun 買い物 + を + する's stem 'し' + purposive に — 'in order to do shopping.' ~かもしれない expresses possibility — 'might, may, perhaps.' Casual contractions: ~かも.