Haven't tried this yet, but I believe Excel 2013 only gives you one worksheet by default, so instead of just switching to application.Workbooks(1).Worksheets(2) you'll need to create the worksheets first. Should be something like

set dpSheet = application.Workbooks.add

Finally got this to work - have to add sheet to workbook differently because Excel 2013 starts with only one worksheet in a workbook by default instead of 3.  Because I have users that have earlier versions of Excel as well I have to check on version as well.   Excel 2007 is Version 12.0   Excel 2013 is Version 15.0.

Set application = CreateObject("Excel.Application")
    appver = application.version
    application.Visible = True
    
    If application.version <= "12.0"  Then
        application.Workbooks.Add
        application.Workbooks(1).Activate

    else
        Set workbook = application.Workbooks.Add
        workbook.activate
        Set colSheets = Workbook.Sheets
        colSheets.Add ,,2
    End If
                
    Set sheet = application.Workbooks(1).Worksheets(1)
        
    If achoice = "All Reports" Or achoice = "Out of Date" Then
        
        Set sheet = application.Workbooks(1).ActiveSheet
        sheet.Name = "Overdue"
    
        Set dpSheet = application.Workbooks(1).Worksheets(2)
        Set dpaSheet = application.Workbooks(1).Worksheets(3)
        dpSheet.Name = "Approval Started"
        dpaSheet.Name = "All Docs"
            
    End If