The answer to your specific question, is NO.
It is possible to do however, I just said with your situation using stock Raspbian it is not possible. Sorry it has to come out like that but in your situation you are pretty much, in a situation that needs some thought now.
In order to rewrite the SD card you need to be able to boot the device into a "bootloader" type pre-environment. This pre environment can be something very simple like busy box, that would normally daisy chain loading the next OS on the SD card.
But when you need to upgrade your OS you trigger the bootloader to download a new image (so it needs networking too) and then a script to specifically overwrite a specific partition, with the correct config....
Yes.. its a load to take in but maybe your company should dedicate some time and think about this situation now before it gets worse.
Good Luck! :)