Atif is partially correct here - Magento does not come with this by default, however you can purchase an extension like WebForms by MageMe and do a little extra footwork to make this happen.
Using this extension and some smart coding, you have the ability to "nest" forms within each other. I would start with one top level form that has a dropdown to select which department the email should go to, then from there set up HTML blocks for each one of these departments. Using the Logic function provided in Webforms, set each block to only be visible when that specific item in the dropdown is selected, and by default hide all of them. This will require the customer to make a selection before being able to submit any information.
Your next steps would be to create a form for each of these departments and when finished, use the widget short-code provided to reference each of these forms in their top level html block.
The main issue i can see with this is that WebForms will automatically add the submit button to the form, instead of asking you where to place it. You might be able to change a template for base layout to use on the top level form since technically it will not have it's own submission.
I hope that helps to put you in the right direction of a way you might be able to make this happen!