What makes touch lamps work? When they no longer turn on and off with a touch, how easy is it to repair? Are these parts generally available at most major hardware stores?

These devices use what is know as a "Capacitance Switch". When you touch the lamp, your body changes the capacitance of the circuit causing it to "Change State" tripping to the next operation. This could be On/Off or Bright/Dim. Because of the size of these circuits, they are impossible to repair. The best thing is to replace.

Here are a couple of items that I have found causing problems with these switches.
First is dirt or oils on the metal surface that you touch. Try cleaning.

Next, if you electrically handy, you could check all the connection with attention to the connect for the touch surface. Bad connection here and the circuit won’t "See" your touch.

At lastly is air humidity. To dry, and your body capacitance will very low when you touch. Try this, dampen your hard and touch the lamp. Not soaking wet, just as if you had licked it or as damp as after you wash and dried your hands.

Other then that, I can not see any help next to replacing it. I have not seen, which does not mean not available, replacement units that you could just change out with out buying a whole new lamp. But I have not looked.

I hope this help and if anyone has anything to add, please do. I just might learn something myself.

Just one last note. Be sure you use only the correct size bulb. To high of wattage will cause it to fail. To small, they may not work very well.

I have a unit that I tried to get to work with a 7 watt nightligh bulb. Not enough load for it to operate. Minimum for that device was 20 watt. This was not a touch switch.

WoW, people show me where the stones are.