。
She went to no end of trouble to find them a house.
Literal
She [topic-は] their [genitive-の] sake [for-ために] effort [object-を] not-sparing house [object-を] find-did-for.
Three set elements stack here. ~ために ('for the sake of') marks the beneficiary. 労をいとわず ('sparing no effort') is a fixed expression — 労 ('labor, pains') + を + いとわず (negative continuative of 厭う 'be loath to, balk at') = 'without balking at the effort.' Slightly literary, often used to praise generous behavior. The benefactive auxiliary ~てやる adds the 'doing-for-them' nuance to 見つけて ('find'). Note that やる (vs. あげる) here doesn't sound condescending — it's the casual default when the recipient is younger, lower in status, or an in-group member.