I don't really know how to do this and there may well be a better way but,
hopefully, this will put you on the right track.

After saving the file as HTML, close it, read it as text and apply it to the
e-mail's HTMLBody property. You may  have to make sure Outlook is set up
with HTML format as default - and I don't know how to do that
programmatically. Anyway here's some amended code that may work - let me
know.

   FileName = "C:\whatever.htm"

   ActiveDocument.SaveAs FileName, wdFormatHTML
   ActiveDocument.Close

   Set fso = CreateObject("Scripting.FileSystemObject")
   Set objHTML = fso.GetFile(FileName).OpenAsTextStream(1, -2)
   strHTML = objHTML.ReadAll
   objHTML.Close
   Set objHTML = Nothing
   Set fso = Nothing

   Dim appOL 'As Outlook.Application
   Dim E_Mail 'As Outlook.MailItem
   Dim Needed 'As Outlook.Inspector

   Set appOL = CreateObject("Outlook.Application")
   Set E_Mail = appOL.CreateItem(0) 'olMailItem)
   Set Needed = E_Mail.GetInspector
   E_Mail.Recipients.Add "User@Example.com"
   E_Mail.Subject = "Something meaningful"
   'E_Mail.Attachments.Add whatever_you_saved_it_as
   E_Mail.HTMLBody = strHTML
   E_Mail.Send

   Set Needed = Nothing
   Set E_Mail = Nothing
   Set appOL = Nothing