Which sentence is grammatically correct:
The statute stated that,"The "vegan..."
The statue stated,"The vegan.."
Do you add the word that or not?
Words like state and say take two similar but not identical structures as their object:
They can take a quoted string, the words uttered: He said "I won't go". This is sometimes called "direct speech".
Or they can take a clause introduced by that. This is "reported speech", and pronouns are substituted, and the verb are backshifted: He said that he wouldn't go.
That can often be omitted in reported speech; but it can never be inserted in direct speech.