Yes, it is possible with COM based add-ins (VSTO based ones).
The Application.NewMailEx event fires once for every received item that is processed by Microsoft Outlook. The item can be one of several different item types, for example, MailItem, MeetingItem, or SharingItem. The EntryIDsCollection string contains the Entry ID that corresponds to that item. Use the Entry ID specified in the EntryIDCollection string to call the NameSpace.GetItemFromID method and process the item.
If you need to handle any specific folder in Outlook and process only items dropped to the folder you may use the Items.ItemAdd event which is fired when one or more items are added to the specified collection. Note, this event does not run when a large number of items are added to the folder at once (more than sixteen).
You may consider your VSTO add-in as a regular windows .net application. So, standard mechanisms can be used to make a web call. For example, you may find the Calling Web Services with HttpWebRequest, WebClient and HttpClient article helpful.
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