Yes — through the office path. You must either reside in New York or maintain an office or place of business in the state. A non-resident with a Manhattan office qualifies through that office, and the office’s county becomes the county of qualification — it appears on every notarial certificate and never changes, even if you later move. Otherwise the standard requirements apply: good moral character, the equivalent of a common school education, and familiarity with a notary’s duties (that last one is what the exam demonstrates).
