Does the iTunes Affiliate Program work with the Volume Purchase Program? Can one earn affiliate commission through the iTunes Affiliate Program for purchases made through Apple's Volume Purchase Programs for Business or Education?

In case it makes a difference, we're based in the U.S., so we would be using the LinkShare-powered program.
That would be called Double dipping.

As Volume purchasing (authorized) user you get the benefits of Volume purchase.

Means, you have to register with Volume Purchase account, that is not the same as the associate account.

You can not get at the same time the affiliate commission on top.