The police report has indicated that the brakes were indeed faulty. Where should I go from there?

You should speak with an attorney a soon as you can. Since the accident happened in NJ, NJ law applies, including NJ's no fault rules. However, you can bring a claim in NY since you live here, and it sounds like you and/or your sister rented the vehicle here. However, some more details would be needed.

The car was actually rented upstate in Montecillo. Would that cause an issue?

No. It shouldn't make a difference since the rental car company is in NY and conducts business in NY. Who rented the car? do you win a car or live with someone that does?