Which part of my autofilter code doesn't work in 2003?

Everything after the first line - there is no Sort object in 2003. You need to use the range's Sort method - seomthing like:

With ActiveWorkbook.Worksheets("BranchList")
.Range("B6:AE65513").Sort Key1:=.Range("E6"), Order1:=xlAscending, Header:=xlNo, _
        OrderCustom:=1, MatchCase:=False, Orientation:=xlTopToBottom, _
        DataOption1:=xlSortNormal
End With