<p>How would I go about implementing dynamic/docking splitter panes in a vb.net app similar to visual studio?</p>

Not easily is the short answer.
A high level idea would be to define some regions using divs or a table and using your js framework of choice make these elements resizeable.   that gets you the splitter aspect.
The docking will have to use absoluting positioned elements that you can drag and drop and if you are currently over a docking element, reposition the element to be docked to inside the docking element and change it's position back to normal.  When you want to drag it out you will change it's positioning back to absolute
Also, this is not easy to do and will take a long time and still probably not work correctly.  Sorry to sound pessimistic though.