Is this acceptable usage when frustrated with a web page: "It won't click"

Solution 1:

It won't click is playfully colloquial, but I think it's fine. It's almost as if you're making the virtual button into a physical object. If it were a physical button, you'd say "it won't press." In that case, you would literally be saying that the button wasn't depressing. It won't click plays on that.

My first choice would be it won't respond. This would work for normal links or other buttons. Also, it won't recognize the click could work.

Solution 2:

I think a more useful phrase for the developer, and one that would be plausibly plain-English enough to come out of the mouth of an end-user would be this:

"Nothing happens when I click the link/button."