In CA I wouldn't be surprised if you needed to have it to pass. That said, I'm no authority on CA emissions- or emissions in general. But if you fill in the blanks (what type/make/year of the vehicle, engine size, etc.) and maybe a photo or two, we might be able to help.
But if you don't have a place on the exhaust manifold to pick up the heated air, the air filter housing might not even be original to that vehicle.
Generally, the vacuum source should be controlled by a temperature activated vacuum port, so the flap is closed during warm up (allowing warmed air to the carb) and open after it's warm. Usually the default setting (no vacuum applied) would be to have the flap open to allow cold air into the carb, but that's not always the case. If you look at how yours is working, you will see it is either always open or always closed with the engine running (and vacuum going to the vacuum motor on the snorkel). By disconnecting the vacuum line to it w/the engine running, then reconnecting it you can see if it's working at all. If the flapper is always closed when connected to the vacuum port (engine running), disconnect it to leave the flapper open.
Performance-wise, you can go w/o the thermo air cleaner assembly, providing the flap is open while the engine is running, and provided you're not driving in arctic conditions. But the smog police may say otherwise.