" It is impossible to fish another cable into the outlet below the sink."

Why?
It may take some work but fishing in existing construction is a pretty routine thing.

I tried fishing but there isn't any room to maneuver. All I do is hit a stud. Our kitchen was just remodeled, so the sink cabinet is in the way of the outlet box underneath. The existing cable is stapled somewhere on the stud, so I can't even use that to pull an additional 3-wire through. There must be a way to re-wire the existing electrical, isn't there?

You can add a switch under the counter that connects to the receptacle or add an air switch to operate the disposal.

Thanks normel. I've got everything disconnected, and I'm not remembering exactly how everything goes. I've got 2 - 12/2 romex coming in to the duplex box, and 1 - 12/2 romex going out to the receptacle under the sink. I do remember a black jumper going to the switch from the outlet in the duplex (possibly a white jumper as well), and also I believe some of the black wires were tied together. Other than that, I'm not having much luck. Also, do I need to split the circuit at the receptacle under the sink when adding a switch? Thanks for your help.

"All I do is hit a stud."

That is why electricians own 24 inch long drill bits, and even 60 inch flexable drills (that can be extended even longer) along with all sorts of right angle drills to work in tight places.

New work is required to be stapled at each and every box.
Fished cables are exempt from this requirement.

If you are willing to make do with the existing cables you may be able to make something work, but it sounds like the power feed may be in the switch box.
The only thing you could do without another cable (or a cable with more conductors) is to use the line from the counter as a feed and install a new switch under the counter for the GD.

It's none of my business and I'm probably wrong, but I'll stick my nose in here anyway.

I have been under the impression for a long time that a dishwasher is required to be on a separate circuit of it's own. When I installed our first dishwasher 10 years ago I ran 14/2 w/gr Romex from an unused 15A breaker in the entrance box across the basement joists and up to the dishwasher terminals according to instructions that came with the appliance. I could have saved myself some time and work by just tying into an existing 12awg 20A circuit that serves several receptacles over the countertop, but I thought it had to be on it's on circuit.
Oh well, I suppose it's better like this since those recepts are on a GFCI protected circuit and I would probably be resetting that gadget every time we wash dishes.

emp jr you are correct in a way, its recomended by most makers that the dishwasher be on its own 15 or 20 amp circuit and that is the most comman practice for most installs. in the install manual for the whirpool I was looking to get it stated a seprate circuit was recomended not required. Most new dishwashers use less power than the ones made years ago so if you had a 20 circuit that was never under much load it would still work well.

I believe the countertop circuits are required to be on their own circuit. The dishwasher is not but cannot be on thhe countertop circuits.