、、。
Let's ask the neighbors to look after the dog while we're away.
Literal
Next [of-の] people [to-に], our [topic-は] absent [of-の] during [object-を], dog [of-の] care [object-を] do [favor-てくれる] [so that-ように] request-[let's-ましょう].
~てくれるように頼む in the volitional form ~頼みましょう ('let's ask') — 'ask someone to do X for us as a favor.' 世話をする ('to take care of') is a set verb phrase. 留守 ('absent, away from home') is a core Japanese household vocabulary word. 間 ('during, while') marks the temporal scope.