I'm using a rich web application that replaces my browser's right-click context menu with a menu of custom options. I want to be able to open a link in a new tab and right-click doesn't work for the aforementioned reason. CTRL + click does not work either. Is there a way to issue a key command that will perform some kind of action on whatever link happens to be under the cursor?
In the case of AutoHotkey the answer would be no. You wouldn't have easy access to the underlying information contained in the browser window and you can't tell the mouse to click something (i.e., automate an existing interface) because there's nothing to click or right-click/app menu/whatever.
You fail to mention which browser you're using... one option you may be able to look into would be the use of a Greasemonkey script in Firefox, assuming you can (or would) use Firefox. Greasemonkey gives you a way to modify the web page content before or after it loads so you can change links within a certain context (in this case the context would be URLs associated with your rich web app), modify the web pages that are loading, etc.