English Translation
Everyone is more or less conceited.
大なり = おおなり but 小なり = しょうなり？why is 大 read.in 訓読み but 小 read in 音読み？
大なり小なり should be pronounced as だいなりしょうなり.
By the way, 大なり小なり sounds unnatural a bit in this sentence. My alternative translation is:
誰にでも多少の自惚れはある。（だれにでもたしょうのじぼれはある）
Below is my own opinion and may not be widely agreeable among other native speakers.
多少の describes the amount or degree of a certain condition.
大なり小なり is frequently used when the speaker concludes that “there is no significant difference”. For example, 誰にでも大なり小なり夢はある = Everyone has a sort of dream. Some have big dreams, and some do small ones. I don’t care about the scale, but my point is everyone has one.
自惚れ is a type of personality or condition.
夢 as a noun is not a state of mind or condition. If you replace 夢 with 夢見がちなところ (a dreamy personality), you can treat it the same as 自惚れ, and thereby 多少の works as a proper collocation.