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We had someone take our picture on the beach.
Literal
We [topic-は] beach [at-で] photo [object-を] take-[have-done-てもらった].
A standard benefactive construction: 撮ってもらう = 'have (someone) take (a photo for us).' Here in past form 撮ってもらった. で marks the location of the action (the beach). 写真を撮る ('take a photo') is the standard verb phrase for photography. This kind of request-benefactive is extremely common in tourist and travel contexts.