I decided to go ahead and test this. I downloaded dokuwiki on a stick. Unzipped the file and copied the directory over to the network drive. I ran mapache.exe and I was able to login as admin and create an new user. From a separate computer my coworker also ran mapache and could login as the new user.
We both could access the file. Dokuwiki locks the page when one user clicks (edit). But the second user could still edit other pages that were not being edited by the first user.
So this seems like it is exactly what I am looking for, a local personal wiki that can be handle multiple users. However I deleted it for now because I need to contact the sysadmin to make sure that I'm not misusing network resources.
Is this relatively secure? I want the wiki to only be accessible to users with access to the network drive. I think it is because I can only launch mapache if I have access to the network drive and then the pages are accessible only through http://localhost:8800/
Unfortunately we all have real world IP addresses so I'm still slightly concerned about security.
All comments are greatly appreciated. This is the first portable wiki I've tried that allows multiple users and page locking during edits.