I have a Maytag 25 Cubic Ft refrigerator freezer on the bottom.
Model # MFD 2561 HEQ
The ice maker stopped working but the water dispenser works fine.
Could there be a clog in the water system that is causing this problem or do I have a ice maker problem that will need replacement?

What you probably have is a clog or bad solenoid in the water valve for the ice maker. Make a jumper test line with an old electrical cord and two push on solderless crimp-on connecters that match the type supplying power to the water valve. Disconnect water line leaving the water valve and attach the jumper test line to the water valve. Plug in the line and see if water comes out. You should hear the solenoid energize. If nothing happens you may have a bad solenoid or a stopped up water valve. Remove it and test the solenoid with an ohm meter which should read about from 700 to 2000 ohms or so. If it reads infinity (open) it's bad. Next take the valve apart and soak the diaphragm and pin in Lime-Away or vinegar until all lime is gone. Clean any screen on the inlet, too. Reassemble and test again. It should be fixed, but make sure to test it. While you have the back off your refrigerator make sure to clean the coils and fan with spray soap and water (a hose nozzle works best). Dirty coils will keep it from making ice, too. Here are some manuals:
https://dl.dropboxusercontent.com/u/8010163/How%20to%20use%20Test%20Equipment%20Service%20Manu al.pdf
https://dl.dropboxusercontent.com/u/8010163/Maytag%20Refrigerators.zip
https://dl.dropboxusercontent.com/u/8010163/Whirlpool%20%20Icemaker%20Ice%20Dispenser.pdf