Maybe "报复" could be better in Chinese,such as 我要报复我的邻居，因为他放音乐太大声了‘’ 。 复仇 is not connected with anything followed,But 报复 can be followed by somebody.

Hi Yair! I would say 我对我的邻居复仇了 or 我对我的邻居报仇了。 Literally, both of these say "I to my neighbor took revenge." There might be a better way to say this, although it's not coming to me at this time because my mind is usually not focused on revenge =D

报复 邻居受到了我的报复是因为他播放音乐的声音太大了。

Actually，we Chinese usually use the word "复仇" as an intransitive verb.Eg."我要复仇！" If you want to say" take revenge on sb. " you can use the word " 报复" cause this is a transitive verb. and “我要报复我的邻居因为他音乐声放得很大” means “I will take revenge on my neighbor for putting on the music really loud”