Our site will contain a link to an external application. This application will need to verify the user is currently logged into our Moodle site. We will not be sharing password information, purely a notification that they are logged into Moodle. We will also provide some userinfo fields to this external application. I am wondering the best way to accomplish this.
If you can, get the other system to implement the IMS LTI standard (http://www.imsglobal.org/toolsinteroperability2.cfm). Then you can easily add it to Moodle (or any other similar system) as an 'External tool'.
Unfortunately, I do not believe this will be an option. Any other ideas?
You should consider implementing CAS support in external app. Than setup CAS server and hook both moodle and external system to it. That way you will have single-sign-on between both systems since Moodle ships with CAS plugin.
CAS site
http://jasig.github.io/cas/4.0.x/index.html