Yes you can. You will need a 4-channel line level converter such as this device.
Scosche SLC4 Four-channel line output converter at Crutchfield.com
You just wire the speaker outputs from the stock stereo into this device and it will convert signal to line level and you can use rca patch cables to your amp.
I would ditch the rear speakers and power channels 1-2 to your fronts. Install some good component speakers. Then the bridge the rear remaining channels for a subwoofer.
You could have a nice system for a few hundred bucks.