As you (probably) know, ～なる means "to become". For ナ-adjectives and nouns, it is preceded by に, such as 友達になる, 元気になる, etc. For イ-adjectives and words that conjugate like them, it is preceded by the "adverbial" ～く form; 新しくなる, 明るくなる, 行きたくなる ("become wanting to go"), etc.
This is just a case of the ～ない form of the verbー取れない (which conjugates like an イ-adj.)ーbehaving "adverbially" in front of なる. So it would translate as, "A Tokyo woman became unable (取れる is the potential form) to be reached after 2017/10/23, and her whereabouts were unknown".
Also, would that be different if I use 「連絡が取れない」?
You could not just use 取れない here because of the から in the sentence. It's saying that starting on that date, contact with her could not be made, and continued on for some time. If it were に instead of から, then you could use 取れない because then it's just a one-time action.