How do you install an universal third brake light on a 2003 Toyota Camry when there is no third brake light to replace?

You mount it to the rear package shelf, find the proper wire and connect. Be careful or you will find it blinking when the turn signals are used.

This must have had a rear spoiler at on time...
The harness for the Third light should be under a cover on the rear parcel shelf. You may have to find it thru the trunk and stick it up from there where you can reach it from inside.
Personally, I'd go to the salvage yard and find an original equipment Third.