Assuming those are working drawers then I am guessing you will have issues. If you have your vessel offset to either side above one of the cabinet doors AND
the wall is set up so that the drainage connection and water connection are both within that portion of the buffet then it will be fairly straight forward. But unless you are setting up your plumbing specifically for this project and don't mind the vessel being offset, I am guessing the drawers are going to be in the way.
You can probably alter the drawers in a way that can accommodate your bathroom but that will be the most difficult part. Everything else is novice DIYer stuff.
Also do you already have this buffet? If your plan was to buy this new, I would suggest contacting Matt Hillard (I found him through this forum) about a custom vanity. I didn't end up using him because my budget was completely blown on other projects, but his price was significantly better than other custom cabinet companies. Here is a link to the thread with his contact info and some work he did. http://www.modernphoenix.net/forums/viewtopic.php?f=5&t=4400&p=22201
I will suggest (for any small countertop fabrication) Mesa Fully Formed. Their remnant yard in Gilbert is amazing. Also Terra firma surfaces has a good selection of modern remnants (but for very small projects). The remnant price does not include measuring or install but I have two vanity tops that were fabricated to my drawings exactly and were easily install by myself.