Hello cheddarthief,

Here is the revised macro. It uses cell "A1" as the TO address. You can change this to whichever cell you will be using.
Sub MailSheet()

  Dim FileName As String
  Dim olApp As Object
  Dim shtName As String
  
    shtName = ActiveSheet.Name
    
    FileName = Application.GetSaveAsFilename("Request for Quote " & shtName, "Microsoft Excel File, *.xls")
    If FileName = "False" Then Exit Sub
    
    ActiveSheet.Copy
    ActiveWorkbook.SaveAs FileName
    
    Set olApp = CreateObject("Outlook.Application")
    
      With olApp.CreateItem(0)
        .To = Range("A1")
        .Subject = "Request for Quote"
        .Body = ""
        .Attachments.Add FileName, 1
        .Display
      End With
    
Cleanup:
      ActiveWorkbook.Close False
      Kill FileName
    
End Sub