I finally got this working. I had found some documentation about changing the port number, but could not get that to work. I had also been trying some other Kee related browser pluggins. I tried this on a new fresh machine and was able to get it to work.
What you need to do is go into the KeePassRPC plug-in in KeePass and change the port number (make it different for each user). Then change the port number in the Kee browser extension (under options). Each user must have a different port. Note that the Kee browser extension has to match the KeePassRPC plugin port for that user. Then revoke any existing connections. Sign Out form that user and log back in as that user. Log into KeePass then start up the browser. it should prompt to make a new connection. I was able to make this work running the Kee extension in Firefox and Chrome with multiple users.