Security light

If the vehicle is starting and running, then the theft light remaining on means there is an issue with the system but not with sending and receiving the security code.

This vehicle has the pellet style system which is basically just a resistor in the key that tells the theft system that it is a valid key.

What I would suggest is we start with checking for all codes. Even though you don't have a check engine light on, you should still have a body or B code in the system that will tell us what the issue is.

https://www.2carpros.com/articles/checking-a-service-engine-soon-or-check-engine-light-on-or-flashing

I am going to guess we have an issue with the remote function actuator module, but I am attaching the wiring diagrams below so that we can check the voltage at each of the switches on the last attachment.

https://www.2carpros.com/articles/how-to-check-wiring

Please run through this and let me know what you find. Thanks