The real issue is how to interface a PICAXE to USB, particularly difficult because the PICAXE would have to be a master and the USB device the slave, and most interface chips ( such as from FTDI ) are designed to create slave devices which connect to a master, not the other way round.
It's technically possible - A PICAXE could talk serially to a PC which stores data on the USB device - but I don't know of anything off-the-shelf which could do that which isn't a PC.
USB Host chipsets are available, but I suspect that they are not very suitable for connection to PICAXE, and the USB protocol isn't that simple to handle.