You may be interpreting what an application like Internet Explorer does for you as a general "feature" of textboxes (a.k.a. string controls in LabVIEW). This is not a general feature of the controls, even with the standard textbox that is available in Visual Studio. As tst mentioned, the System Combobox is the control you want.
If you must use a textbox (i.e., you don't want to display a dropdown), you then have to program this feature yourself. Note that tst's example doesn't quite work right in LabVIEW 8, as the scrollbar is still visible after the table is dismissed. Still, the basic idea is to respond to an event with the control and make another control visible, which in your case could be a listbox, for example. The even that you would respond to is a key press event.