After installing the WebCam SDK for Logitech cameras and the logitech
webcam, I try to run the demo programs that come with the software but
none of them can run because they all complain about being unable to
detect the camera.
What is wrong?

This could be a number of problems so I am going to suggest some
possible problems and solutions.

1. The camera cable is loose or not connected. (If your camera has a
power connection check that as well)

2. Try leaving the camera connected and ON when you start your
computer. If you get a "new hardware found" box pop up (in windows of
course) then it should install the drivers for your camera if you have
installed the software that came with the camera.

3. If the camera does not connect by USB (small black cable with
rectangular connector) then the program that connects to the camera
will have to be told which communications (commport) port to use. The
port will be com 1 to com 5 and if one doesn't work you can just try
them all.

If any of these suggestions work (or more likely don't work) please
write back to let others know so they'll have a better assessment of
the problem.

Good luck, device troubleshooting is tricky sometimes but it is
usually a problem that can be fixed.

The Logitech QuickCam I've used has two cables. One plugs into your
mouse port (PS2), the other plugs into the LPT port. The mouse has to
be inserted into another female plug attached to the camera cable.
When I installed the QuickCam driver, it mentioned that the camera
works only with the LPT version on NT machines.
Since you mention the SDK, you probably want to develop something for
the camera yourself. If you havent already done so, I'd recommend
uninstalling the SDK and installing the actual camera drivers and
software (from a user's perspective) to see if the camera is detected
at all.
If the SDK is meant to work for many models/configurations of the
logitech brand range, the problem might be just a selection of the
right dlls/libraries/headers/linking options etc.
I dont know if this solves your problem or not, but it might help.
Goodluck. 
Being a developer myself, I know how it feels when you're stuck on
some trivial bottleneck problems.