What makes a flapping noise in the engine when car is started/restarted? It sounds like a dull slam of a fabric covered lid, slamming shut!
This car is new, and under warranty. You should be asking your local Toyota dealer to explain the noise.
Yes, the dealer should have to answer that question. But it does sound like the “blend door”, which is a plastic flap inside your dashboard that controls the mixing of hot and cold air inside the heater. Next time you start the car, have a passenger (or you) put his/her ear on the dashboard. If the sound is definitely coming from inside the dash, it’s the blend door. Your dealer will have to figure out why it’s loose.
Difficult to say especially not knowing what make model or year car you have. The number of miles on you car might also be helpful Right now I would guess it is a vacuum control of a heater control that you are hearing.