live (in)

Yes, without "in" the sentence means the house is needed for her to survive, to keep her from dying.

If it was a formal circusmstance (academic paper ), I'd write it as: Susan moved to a new city and found a new house in which she will live."

The good grammar rule is not to end a sentence with a preposition. However, in casual conversation what you wrote is acceptable. In is needed somewhere, to leave it out is improper usage and changes the meaning.