It is possible, of course, you can make a program/script and a always consider that when you boot you will immediately mount the SD card and the USB stick, then you will copy all the photos from the card to the stick.
Of course there are some issues. You will have to find a way to tell you if there are any errors, if the process goes on ok. You can always take a HDMI cable with you and try to find a TV somewhere to plug your Pi and follow the procedure on screen, or you can make a simple LED status system connected directly to the GPIO that will flash a red LED if something goes wrong, and a green one if all worked fine.
How should be that script? First try to mount both the stick and the card, and flash the green LED if you get a positive result to both mounts. Your devices when the kernel boots will most likely be the same from boot to boot, if nothing else changes. So connect them and find out which devices appear. Start your script by mounting them both and examining the error codes of mount (check the manual). Output your GPIO connected LED with a single green flash. (There are some tutorials on how to connect a LED to the Pi online).
Second step you make an inventory of all the pictures in the card and save their names to a file. All the cameras I know name files sequentially, they do not repeat a name if you don't manually reset the counter. Check if your camera is like this to make sure, that should work on your favor.
You can create a new directory on the stick with the current date (even if a offline Pi will not have the correct date, it will provide you with a unique answer, and sequential), or just copy the new files to a unique directory, given that there shouldn't be repeated names.
Once you copy all the files, check for the command's error code and see if it was successful or not. I would not recommend a -y or force, given the value of the data you have, you should manually inspect in case of any errors. Flash the green LED twice if no error was reported. Make the red LED solid if there was some error code that was not success.
Third and last step: compare the manifest you made when the script started and check if every file on that list exists on the destination directory on the USB stick. You can even make some MD5 or CRC test to make sure that there was no corruption in the process, but that will take a while to check. Nothing that a good night of sleep while your Pi works won't solve (I think! Check your timing if you want to ado that!).
If all files are checked according to the manifest, make the green LED solid and enjoy! I don't recommend your script formatting or erasing the card, just do that on the camera manually after the green solid LED.
You can always create a cumulative manifest file and check it at boot time, to make sure nothing happened to your stick while you slept. :)