Pop-up calendar option does not display when I add Date field in webform
The Popup Calendar option in Webform is entirely separate from the Date module, so it should always work regardless of whether you have installed Date module (or Date Popup module).

Make sure you've added a date component to your Webform, then under "Display" there is a checkbox for "Enable popup calendar". This option is enabled by default. Now a little calendar icon should appear next to your date component when viewing the node.