I encountered the same problem. I cannot launch any vehicle from the VAB directly. My current workaround is to save them, go manually to the launchpad from the main menu and select the saved vehicle.
When the bug happens from the VAB, the loading screen image for the launchpad stays visible but there is no loading bar or any loading text displayed. It feels like the loading is complete but the image is not hidden. As ClariNate mentions I can still hear some sounds from moving the mouse over invisible ui elements.