The whole sentence is: 私となりに住んでおります綾瀬と申します
It's from the manga yotsubato, that I'm trying to read along with a translated version. The translated one says "I live right next door. We're the Ayase residence"
So I couldn't find anything conclusive about this なり word, and I also don't know about this と, if it's part of the same expression, or a separate particle, and if it is, I don't know the role it's playing here. Could you please clarify?