Do I have to build an "ASP.net Web Application" to submit HTTPS
requests?  Or can I use a "Windows Application"?

You can use the HTTPClient class in a windows application to simulate
browser requests.

take a look at this link:
http://support.microsoft.com/default.aspx?scid=kb;en-us;303436#appliesto

thats in csharp, but it shouldnt be hard to convert the code.

Windows Application

Invalid Cast Exception using Inherited Object
DateTimePicker - Howto set the language
String Concatenation
PC to PC file transfer
Save and Load a TreeView to SQL Database
Referencing the Assembly GUID
Client- Server Programming using VB.NEt & SQL SERVER
How to hide the dialog form?
Limit Columns shown from XML file
OLE Container