Apalis TK1: Flash Linux via USB

Is it possible to only use the USB connection to flash everything from recovery mode?
I want to integrate the Apalis TK1 module and I have the USB recovery port and debug serial port available.
The flashing method using the SD card is not suitable for me.

I know this is possible on the NVIDIA Jetson TK1 development kit.

Sure, why not? While we do not support any such out-of-the-box by any of our update scripts one could use U-Boot’s UMS feature e.g. as explained here.