No you can't.
The way audio is recorded onto vinyl at the pressing stage removes or 'attenuates' the bass frequencies. So if you record direct to your soundcard it will record something, but it will be so quiet and tinny that it would crap to listen to. That's why you need an amp.
Get down to Maplin/Dixons whoever and buy a long phono lead. Check your soundcard line-in socket.
These are usually 3.5mm Stereo Jacks so you may need a 'Y' Phono lead. One end has one red and one white phono jacks - the other has what looks like Walkman Headphone jack. Shouldn't cost anymore than a fiver.