I think Articles does this, but if CustomRequest doesn't do what you want, you'd need a custom snippet to parse the URL.
I'm not sure there's a way to make a link tag create the URL you want, though. This might work is you set the URL suffix to empty:
The problem is that MODX wouldn't find the file (since no resource with that alias exists).
I think this would work (with a custom snippet). You'd have to send everyone to the same page and have the custom snippet forward them on.
You could also use a form to send a "Post" request, so the search term wouldn't end up in the URL at all.