There isn't direct exposure of the Copy and Paste functionality on the ComponentOccurrence object, which is the API object that represents each of the nodes in the assembly tree (parts and subassemblies). However, it is possible to directly call Inventor commands. The code below illustrates this.
Public Sub CopyPaste()
Dim asmDoc As AssemblyDocument
Set asmDoc = ThisApplication.ActiveDocument
Dim occ As ComponentOccurrence
Set occ = asmDoc.ComponentDefinition.Occurrences.ItemByName("L:1")
' Get references to the copy and paste commands.
Dim controlDefs As ControlDefinitions
Set controlDefs = ThisApplication.CommandManager.ControlDefinitions
Dim copyCommand As ControlDefinition
Set copyCommand = controlDefs.Item("AppCopyCmd")
Dim pasteCommand As ControlDefinition
Set pasteCommand = controlDefs.Item("AppPasteCmd")
' Select the occurrence.
asmDoc.SelectSet.Clear
Call asmDoc.SelectSet.Select(occ)
' Execute the copy command.
copyCommand.Execute
' Execute the paste command.
pasteCommand.Execute
End Sub