Programming Editor supports programming remotely with the
COM > TCP/IP option on the Serial Port tab in the Options dialog. You need to check that 'Show port help tools' checkbox first.
Also, might be relevant,
booti2c command can be used to copy a program from an i2c EEPROM or the
run command to just run a program on the EEPROM. However PICAXE doesn't support multi-mastered i2c (but you could do your own with separate handshaking signals) nor can you generate a program for one on another PICAXE without an extremely long session of reverse engineering of the tokeniser ("compiler") so the programs would have to be pre-generated on a computer.