The correct option is
A We eat not only vegetarian food, but also sea food.
The conjunction, 'Not only.. but also' is a correlative conjunction, which combines two grammatically equal items. This means that the words which follow both parts (i.e 'not only' as well as 'but also') should belong to the same parts of speech. Using different parts of the speech after 'not only' and 'but also' makes the sentence imbalanced and grammatically incorrect.
Option A: The phrase, 'not only' is followed by 'vegetarian food', which is a noun phrase. The phrase 'but also' is followed by 'sea food', which is also, a noun phrase. Hence, this option is grammatically correct.
Option B: The phrase, 'not only' is followed by 'eat' which is a verb. However, the phrase 'but also' is followed by 'sea food', which is a noun phrase. Hence, this option is grammatically incorrect.
Option C: When a sentence begins with 'not only, it is followed by an auxiliary verb like 'did'/does'/'is' + subject. In this option, the sentence begins with 'Not Only'. However, this is not immediately followed by any auxiliary verb. Hence, this option is grammatically incorrect.
Option D: The phrase, 'not only' is followed by 'that', which is a demonstrative pronoun. However, the phrase 'but also' is followed by 'sea food', which is a noun phrase. Hence, this option is grammatically incorrect.