You are probably going to have issues trying to export from an old version of Access to a newer version of Excel
However you could try changing the spreadsheet type in the code from acSpreadsheetTypeExcel9 (or similar) to acSpreadsheetTypeExcel12 (for Excel 2007 or later)
Syntax something like:
Code:
DoCmd.TransferSpreadsheet acExport, acSpreadsheetTypeExcel12, TableName, ExcelFile, True, Range
Can you export to a CSV then open said CSV in excel? The DoCmd.TransferText method may offer another route to achieve the same results.
This should avoid version issues with different users having different versions of excel. Also affords the ability to import into earlier versions, should one still have one.