No applications respond to trackpad clicks
This sounds like a perfect candidate for VoiceOver. It's fairly (if not entirely) straightforward to learn and use.

Hit Command ⌘+F5 to turn on VoiceOver. (You may need to use the fn key as well to hit F5.) You can then hold down Control ⌃+Option ⌥ (usually referred to as the VO keys) and use the arrow keys to navigate around the screen.

Here is Apple's page on VoiceOver and here is a tutorial on how to use the technology.