Andrew,
There is no stock way of doing this. However, you could customize this by adding a File Attachment field to the ExpReportExpenseReport table. Then you'd just have to add that field on to the New/Edit Expense screen (ExpReport_EditExpenseReport). Remember, you'll have to copy the screen first and edit the copy. There may be other screens where you may want to add this field, but I see this as your starting point.
If you'd like help with this type of customization, we have Method Consultants in house that can help you or you can contact a Method Solution Provider.
Hope this helps.
Adam