The best way to know is to see the watch log or history of your visited webpages. You can find which websites were visited and if any was for ordering.
Look at the page history in your web browser. Other than that, there's no hard and fast way to see if your computer was used to order something on the internet; internet sales aren't logged in any standardized way.
Two things: check browser history and cookies which normally have some trace of ordering - then check your paypal and credit card. It may not be possible to tell absolutely.
It's going to be far easier to lookup recent purchases using their/your debit or credit card. Check with your ISP for a list of sites using your IP address.