In the 2003 toyota echo. The right rear socket has no voltage. Maybe a ground wire
Is loose. I wonder how the auto shop is going to find, why the voltage is not
Getting to the socket.(new bulb)( voltage tester) all the rest of the lights work okay.

No voltage coming into the socket contacts. But Why? The right directional signal fuse is in a circuit, where the rest of the lights work.
The bulb too also acts as a hazard light. Which doesn't work either.

First, check all under-dash and under-hood fuses with a test light or multimeter:

https://www.askmehelpdesk.com/cars-trucks/faq-how-troubleshoot-repair-maintain-hondas-46563-3.html#post252145

Next, check for voltage in the socket with a test light or multimeter. Look for obvious corrosion and then start performing continuity checks on the circuit.

Is your indicator relay working?
Check for voltage at the RELAY