ISTM the sound of the fan would scare the cat at least as much as the click.
Why not put it on a timer?

Randy427 - I know it is the click that scares her because I've seen her jump in response to it. The cabinet is in a bathroom which has a ceiling fan, which is also on a motion sensor, and it no longer scares her. I wanted to put the fan in the cabinet on a timer so that it would start a certain amount of time after using the box but I have two cats and there is no guarantee that the fan wouldn't be on when either one tried to use the box - so they have to get used to the fan. I think if I could get rid of the click I could acclimate her to the fan (the fan itself is in the attic and, although it still makes noise, I think she would get used to it because she is used to the ceiling fan). But cause the click puts her on edge to begin with, getting her used to the fan is proving to be a challenge.

I bought this one from HD and really like it - no clicking and a lot of customizable features built in.