Customarily is an adverb, but you have not used it correctly in this sentence. In this case, you have begun the sentence with "it is." It is a pronoun, so it takes an adjective, not an adverb. Here are two equally correct ways to express this idea:
It is CUSTOMARY in many societies for diners not to use their left hand when eating.
In many societies, diners CUSTOMARILY eat with only their right hand.