From looking at a “typical” wiring diagram for the Camry, it looks like the problem could be any of the flasher unit, the hazard switch, or the turn indicators – but most likely, I would think, by the flasher. Rather to my surprise, it looks like there is power at the flasher unit at all times if the ignition is on, but the the flashing and routing of the flashes to the lights is controlled by a second terminal on the flasher.
The flasher unit is alledged to be in a plastic holder on the firewall near the steering column “behind the interior fuse panel” I don’t see a fuse panel in Haynes picture and the nearest fuse panel I’ve encountered on our 1999 Camry is behind the coin holder on the dash to the left of the steering column. There are apparently both four and five pin versions of the flasher, so you possibly need to pull it and count pins before heading off to the parts store. (It’s dark and raining outside, or I’d go look at our car to see if I can find our flasher from that information).