Well, it would be possible, but it would be slower, clunkier, more prone to errors, etc.

As soon as you store some data locally, you have to do a lot of sanity checking, and synchronisation is a pain, and so on.

Personally I think you should have the option of making an 'always offline' character, with no access to the AH, but I think that would involve Blizzard putting too much data on the client side -- which it obviously doesn't want to do.