You can use our Integration feature to have the order data POSTed to the URL of a script at your end whenever an order comes in, so your script could log the data and do whatever else you wish with it:
http://www.e-junkie.com/ej/help.integration.htm
However, we have no method of remote access to your full logs at our end, aside from the View/Download Transaction Log link in your Seller Admin, so if you set up the Integration method I suggested above, the logs you keep at your end would only have data from that day's orders onward. If you need a backlog of order history prior to that point, you might download a copy of your full log via Admin, so you could import that into the logs you're maintaining at your end.