N.Y. Pub. Off. Law § 37
When a judgment shall be rendered by any court convicting an officer of a felony, or of a crime involving a violation of his oath of office, or declaring the election or appointment of any officer to be void, or that the office of any officer has been forfeited or become vacant, the clerk of such court shall give notice thereof to the governor, stating the cause of such conviction or judgment.
Whenever a public officer shall die before the expiration of his term of office, or shall cease to be a resident of the political subdivision of the state or a municipal corporation in which he is required to be a resident as a condition of continuing in the office, the county clerk of the county in which such officer shall have resided immediately prior to such death or removal, shall immediately give notice of such death or removal to the governor. If the governor is not authorized to fill any vacancy of which he shall have notice, he shall forthwith give notice of the existence of such vacancy to the officer or officers, or to the body or board of officers authorized to fill the vacancy, or if such vacancy may be filled by an election, to the officers authorized to give notice of such election.
N.Y. Pub. Off. Law § 37