Sono riuscito a trovare un pò di materiale che, opportunamente adattato, mi ha risolto il problema:
Private Function GetIp()
Dim myWMI As Object, myobj As Object, itm
Set myWMI = GetObject("winmgmts:\\.\root\cimv2")
Set myobj = myWMI.ExecQuery("Select * from Win32_NetworkAdapterConfiguration Where IPEnabled = True")
For Each itm In myobj
MsgBox (itm.IPAddress(0))
Exit Function
Next
End Function