Typically for something like this you would have a property in your ViewModel that would tell the view what to show. You have lots of different options for how you could do this.
One option would be to have a Results property that your list box is bound to. Put the textbox and button in a Grid and bind the Visibility property of the grid to a property that is Visible is there are no results and not visible if there are.
Lots of different ways to do this.