As you can see on the documentation of VBox you can only define minimum, prefered and maximum range, there's not really a way to make it manually resizable.
The only proper solution to solve your problem is to develop your own class to do it, because what you want seems very specific, with your problem description, I don't think use some layouts or panels will do what you exactly want.
I found something that you can use : Dragging to resize a JavaFX Region
This allows you to resize a region, all you have to do after is to put you VBox in this region, but notice in this article that :
Only height resizing is currently implemented.
This code won't work in JavaFX8, you'll have to check the comment to see how it worls in JavaFX8
Hope this helps.