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She walked all over looking for the dress.
Literal
She [topic-は] that dress [object-を] looking-for [for the purpose of-のに] walked-around.
のに here is the purpose-marker — the nominalizer の plus に, meaning 'in order to / for the purpose of'. Identical surface to the contrastive のに ('despite'), with context the only disambiguator: a verb in plain form before のに followed by another verbal predicate signals purpose, while のに at the end of a clause leans concessive. 歩き回る is a compound verb pairing 歩く ('walk') with 回る ('go around, circulate'), giving 'walk all around'. The ~回る suffix is productive: 探し回る ('search all over'), 走り回る ('run around'), 飛び回る ('fly around').