SD cards cannot be shared between masters in operation, and it is unsafe to disconnect the card from the camera with an electronic switch when the camera is not expecting that.
Nor is an ordinary Arduino fast enough to passively snoop the data being written to the card by the camera (while a card will let an Arduino write to it quite slowly, any camera would do so far, far faster)
Nor is an ordinary Arduino able to do much of anything meaningful with the image files produced by a typical camera in real time.
Nor can an ESP8266 while running Arduino-based software; it apparently does have some high-speed I/O capabilities but that would require low-level rather than Arduino-port-based programming.
For all practical purposes, you'd need to power off the camera, remove the card, and place it in the Arduino reader.