Check the console logs and see if there's anything relevant to the failure being logged there.
If there's nothing and solely as a guess, download the 10.6.8 Server combo update (not the delta) and install it.
Thanks for poking me to look at the console logs! That led me to find the error message:
App Store <Error>: CGBitmapContextGetBitsPerComponent: invalid context 0x112f90060
Which in turn pointed me to https://discussions.apple.com/message/15431640#15431640
Renabling Spotlight (see http://osxdaily.com/2011/12/10/disable-or-enable-spotlight-in-mac-os-x-lion/ ) allowed the App Store to work again!
Just why the App Store requires Spotlight to work is beyond me.