Dynamically generate entire website(s)

I have a client currently using WordPress, but they desire more from their site. They want users to be able to browse their website, sign up for a subscription, and ( once the subscription is confirmed ) an entire website ( a slightly customized version of the client’s site ) is generated on the fly. The end result being multiple versions of the ‘base’ website, sharing the same core and modules and such, but customized for the user.

If WordPress can do this, please help me. If WordPress cannot do this, please help guide me to the proper resource(s).

Thank you,
Alex K