Yes, it looks like it is possible, though it appears to be tricky. I found a video here describing how somebody modded their steam controller using an Xbox-one style joystick.
However, be careful - it looks like the process of modding the steam controller joystick using Xbox or PlayStation joysticks can be difficult without the proper tools and experience, especially when it comes to modifying the joystick itself (which WILL NOT fit onto the controller without some modification - the video above recommends using a drill press to make the joystick fit onto the controller, whereas the Reddit thread recommends using a dremel).
Should you decide to proceed, here is a tutorial for how to take apart the controller in more detail than the video above. Good luck, and happy modding!