Yes you can in theory depending on if the powered sub has "line level" inputs. If not, a device can be acquired separately to perform that interface.
The problem is that you are just going to amplify the tinny sound you now have. I think that the problem with the stock system isn't just the small speaker used as a sub, but the entire subwoofer setup, volume, frequency range etc. It would be best to tie into a point after the primary amp or input to the subwoofer amp which ever would be most convenient. You need to tie into both left and right channels and also have the ability to adjust the gain to balance the system and filters to block the higher frequencies.
If you haven’t upgraded the other speakers in the system it is still going to sound bad. The majority of the sound is produced in the midrange with the door speakers primarily responsible. Adding the sub you will have the tinny sound with a bunch of boom boom to go with it.