Grammatically correct, though perhaps not expressing what is intended.
That's the reason why. I chose vegetarian food.
Is two separate statements. The first.
That's the reason why.
Means the same thing as "That's the reason", but the redundant why can add emphasis or stress it's connection to a question that had asked "why?" This sentence only makes sense after a reason has been given.
I chose vegetarian food.
States that you chose such food. It doesn't really relate to the previous sentence, unless perhaps underlining it or adding to it. E.g.:
Why did you not try the speciality?
The meat dishes all had sauces with chilli in them, and I'm allergic to chilli. That's the reason why. I chose vegetarian food.
First we've the explanation, then the "that's the reason why", and then the final sentence adds separate information.
That's the reason why I chose vegetarian food.
Says of the previous statement not just "that's the reason" but "that's the reason for choosing vegetarian food.