Strange problem with my 1994 Suburban

[Kent]

For well over a year, my 1994 Suburban has been making a strange sound. It seems to be getting worse, so I have decided to solicit help. A little difficult to describe, but when I start the engine, and especially when I turn the engine off, there is a loud clicking (best term I can think of) sound that comes from under the dash on the passenger side. I have never timed it, but the clicking can last for about 10 seconds when you start the engine and at least 30 seconds when you turn off the engine. I am thinking it is somehow related to a vacuum of some sort that keeps opening and closing a valve of some sort. Has anyone experienced this situation before and if yes how do I solve it without spending an arm and a leg?
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