Yes. In principle, a camera can read the sensor with the shutter closed and use that as a dark frame, and some cameras effectively already do this as part of long exposure noise reduction (LENR). LENR typically makes a second exposure of the same duration after the main shot and uses that dark frame to subtract fixed-pattern noise and hot pixels.

So the idea is viable, and it’s likely something the camera processor could do if the firmware exposed it. In practice, though, many cameras don’t offer a user-facing option for a shutter-closed dark-frame capture.

Even if the shutter stays closed, using the lens cap is still sensible. It reduces any chance of stray light leaking through and also avoids accidental shutter damage if the camera is pointed at very bright light, such as the sun.