なるべく姉ちゃんにも周りにも迷惑をかけずに、事態の早期解決をなんとか頼むぞ。

、。

I'm asking you to somehow resolve the situation as quickly as possible without causing trouble for sis or the people around us.

Literal

As-much-as-possible sister [to-also-にも] surroundings [to-also-にも] trouble [obj-を] not-give-[without-ずに], situation [gen-の] early-resolution [obj-を] somehow ask [ぞ].

A masculine order/plea. なるべく ('as much as possible') is a common hedging adverb. お姉ちゃんにも ... 周りにも stacks multiple にも for 'to X also, to Y also.' 迷惑をかける ('to cause trouble') is the fixed collocation; here put in negative ずに form (formal 'without'). 事態の早期解決 ('early resolution of the situation') is compact Sino-Japanese formal language. 頼むぞ is the rough masculine request-assertion. The mix of formal vocabulary with rough ending gives it the flavor of a character talking tough about serious business.