1996 Chrysler Grand Voyager Clicking inside vehicle

Heater cores don't make noise. You really don't want to replace one yourself. The entire dash assembly and steering column must be pulled back, and the air conditioning lines have to be disconnected. That requires recovering the refrigerant first. Refrigerant is very dangerous to work with as it can cause blindness and frostbite. Professionals wear gloves, safety glasses, AND a face shield.

It's more likely you're hearing an actuator on the heater box. They are spring-loaded to default to the defrost position when engine vacuum is lost, typically the result of a broken vacuum hose. That's a safety feature. Your feet might freeze, but the windshield will be clear. You might try an experiment. Switch to defrost just before you stop the engine, then see if the noise is still there when you restart the engine hours later. If the noise doesn't occur until you switch to another mode, it's an actuator.