Time capsule web site logging?
The router itself does have DNS caching, but that cannot be accessed without rooting the device (and probably changing a few settings that shouldn't be changed to do logging to disk). Furthermore, an automatic update may reset everything, so I wouldn't go down that path at all. Much better idea would be to set up a DNS on a small linux machine and then configure the Airport DHCP to use your small Linux computer's DNS server as the default. That will persist through updates, etc. All clients connecting will ask the linux machine for the IP address of the website they want to go to. Just log it.

Also, for what it's worth, even if this is your network and these sorts of invasions of privacy are easy to do, I really really REALLY recommend you make everyone (children, etc) aware that this logging is going on. For everyone's sake.