Certainly possible. While I have used both the SRF05 board and several AXE118 boards, I have not used them together. I no longer have the hardware due to the work being done under contract for a client 15 years ago. So I'm a bit rusty on interfacing the SRF05.
When I opened the datasheet for the AXE118, I noticed that document described the use of an earlier PICAXE, the 20M. The 20M2 is a much more flexible and capable chip, with almost all I/O pins being configurable as inputs or outputs. The PICAXE pins on the left all have a resistor array arranged as pull-downs. This may interfere with the operation of the SRF05, so I suggest you don't use those pins on the PICAXE.
My suggestion would be to use one or two of the pins on the right of the PICAXE: B.0 to B.7. However, to access both the input and output capabilities of these pin you will need to wire the SRF05's I/O into the wiring points between the PICAXE and the ULN2803 chip (IC2). You would not use the ULN2803 (or equivalent) chip because it can only be used as an output buffer/driver and not as an input to the PICAXE.