是的，这个why可以省略。关系副词why用于the reason后引导定语从句常可换成that或for which，均可省略。例如：

That's the reason (why, for which, that) he came. 这就是他来的原因。

又如：

The reason she was ill was that she had eaten bad meat. 《现代英汉词典》

The reason I called was to ask about the plans for Saturday. 《朗文当代英语词典》

The reason these cars are so expensive is that they are largely built by hand.《麦克米兰英语词典》