The basic honest answer is 'no' - Web Designer or Designer Pro are not blog creators and do not have any facility to write new content to a database from a web interface allowing your client to add/update.

However, there are ways to emulate a blog - though it takes some setting up and explaining/training for your client.

► Embedding an updatable google doc

Personally (although I created the above for the exercise) I'd not bother.
There are much better and more convenient blogging systems.