I'm not sure what one of your documents looks like when you initiate this code but why don't you take the spaces out before you ever put it in the clipboard?
Code:
Sub Macro18()
Dim oClipboard As DataObject
Dim oRng As Range
'This part gets a person's name from a Word document to the oClipboard.
Selection.EndKey Unit:=wdStory
Selection.MoveUp Unit:=wdLine, Count:=4
Selection.EndKey Unit:=wdLine, Extend:=wdExtend
Selection.MoveLeft Unit:=wdCharacter, Count:=1, Extend:=wdExtend
Set oRng = Selection.Range
oRng.Text = Trim(oRng.Text)
oRng.Copy
Undo 1 'Put the spaces back in the text.
lbl_Exit:
Exit Sub
End Sub