Here are the two sentences.
"That's the reason why she spoke"
"That's the reason she did so."
Which one sounds natural to native speakers?
Thank you very much!
This issue was covered on ELU long ago. There is a school of thought which claims there's something inherently "wrong" with conjoining reason and why in this way, but in practice most native speakers are either unaware or don't care about such pedantry.
I can't think of any specific context where you can't omit why after reason used as a noun meaning cause. Nor do I think it ever makes any significant difference to the meaning - it's just a stylistic choice.
Because of the "pedantic" position linked to above, you might want to avoid using the reason why yourself (it's never necessary). But don't assume just because someone else does use it, their English is "poor" (though I would say it's more a spoken than a careful formal written usage).