You should be able to convert them, however, be prepared to spend quite a while working on it (not that the USB turntable doesn't take a while!).
Firstly, you'll probably have to find an adaptor for a 1/4" jack to a 3.5mm plug that your soundcard uses. Most decent music hardware shops should stock these or you can get them online for as little as a £1. Then you'll want to plug your LP player's output cable into the input plug of your soundcard.
Once you've done that, the fun starts. Now, there are many commercial applications out there to help you convert your LPs. There is also an Open Source program called "Audacity". Basically, you set Audacity up to record from your soundcard and then once you've got the song, convert it into MP3/whatever format you like. But, you have to do this for each track on the LP (unless, of course, you want all your albums in single MP3s).