Depends what you call the body panels, and the type of collision, especially when talking about unibody cars which is the majority produced today.
Here’s what I’d call a “body panel”: the front fenders, the hood, the doors, and the decklid or liftgate. Of these, the only thing that has no significant crash consideration is the fenders. Obviously well-designed doors are important to intrusion avoidance during a broadside collision (hint: North America has much stronger requirements than Europe does for doors). The hood has to be able to not detach and fly through the window and decapitate the passengers.
There’s also the car’s body sides. Much of what you see from the outside looks like a body panel, but in effect it’s a huge structural element of the vehicle on its own. It keeps your roof from crushing in on a rollover; it contributes to lateral intrusion avoidance, and contributes to rigidity along all three axes of the car.
As for whether a car body could be made completely from bronze, let’s see. I quick search indicates that at least several types of bronzes are suitable for stamping applications, which indicates that body structures could be formed. Joining them would be an issue, as mechanic fasteners would have to be used (e.g., rivets), or a completely new resistance spot weld system would have to be developed (too conductive). Since we’re making the structures from bronze, we could fall back to silicon bronze fusion welding, although this has environmental impacts and is expensive.
So far, we can form the parts, and have a means to join them together, then. So, yeah, if we really wanted to, we could build a car body from at least certain types of bronze.
The real question is, why would we want to, and if we really wanted to, would we want to live with all of the sacrifices that would be involved? Weight, safety, and so on.