Electrolysis of brine without a membrane
So if the membrane is removed, will the chlorine gas react with NaOH to form sodium hypochlorite?


Yes.


  Because I've read that the chlorine will instead just react with the hydroxide ion to produce hypochlorite. 


Yes. This is just another way to say the same thing.

As for the sodium, it does not go anywhere. It simply continues to stay there in the solution, like it did before the reaction. It does not even "know" about the reaction, or that it now "belongs" to the hypochlorite; it is just a spectator ion. Since the dissolution, anions and cations go their separate ways and do not "care" much for each other. This is always the case with ionic salts.