I have a 2007 camry and the trunk release cable that opens the trunk from the driver's side has snapped. How do I replace it? Do I just slide the cable in from the trunk?

I managed to open the cover on the trunk but I can't find out how to remove the carpet.

It goes from the trunk latch, down the trunk hinge, into the trunk compartment, through the back seat to the drivers floor board.
Are you prepared to remove all those interior parts? I'ts quite a job.

Do I really have to remove all the interior parts? I was looking at it and wondering if I could attach the new cable to the old one using some tape or something and pulling it all the way to the trunk using the old cable.

I would try that, I do it all the time on boats, alarm wiring.
When connecting the ends of the cables, keep connection as small and snag free as possible.
Tape very good and maybe use a wire lube or slipstik. Pull carefully, and may help if someone feeds at the same time. If you can't pull through, try to be able to pull back out.

If the 2 ends together is too much, maybe pull a rg6 coax, and use that as a fish, using the center conductor AND the Shield.
Good Luck.