。
She decided to go shopping by herself.
Literal
She [topic-は] alone [as-で] shopping [for-に] go [thing-こと] [to-に] did.
一人で uses で as 'in the manner of' — the same で that builds 二人で ('as a pair'), 家族で ('as a family'), みんなで ('all together'). 買物に行く: に here marks the purpose of going (shopping is the aim of the outing), a use distinct from に for destination or location. ~ことにする expresses a personal decision by the subject — contrast ~ことになる ('it has been decided that'), which presents the decision as externally determined.