the reason she is shy

不知道下文是什么.
The reason she is shy.这句话中省略了that一词,she is shy是个定语从句.
The reason why she is shy.这句话也对,只不过是后面一定要接原因.
我给你举个例子让你对比下,就能理解两者的区别.
The reason that she gave up teaching was not ture.
他辞去了老师的职务是真的.就是没有 that 也完整.是一句叙述句
The reason why she gave up teaching was her serious illness.
之所以他辞去老师的职务是由于严重的疾病.是对“原因”的一个补充说明
对“为什么原因的”补充.就要用why

the reason why。。。。is that这种结构

你好楼主，其实，当reason作先行词时，如果定语从句的引导词作状语，那么这个引导词有四种形式：1、why（较正式）；2、for which（较正式）3、that（非正式） 4、不填 （口语化）。所以，您的事例有四种情况是正确的：
1、The reason that she is shy.
2、The reason why she is shy.
3、The reason ...