Why would the reverse light stay on even when the car is off on your 2003 vauxhall vectra?

Always work to the wiring schematic and electrics chapters in the repair manual.

Ask the question, "What did I do last?" It might have bearing on the fault.

If you have a tow bar fitted check that the connector wiring is correct and that it isn't shored out.

Check the light clusters for bad wiring, especially if there is a tow bar, or reversing sensors are fitted.

Older vehicles had a fixed 12v feed to the lamp and 'switched the ground'. Meaning that the lamp was live and closing switch connected the lamp negative wire to ground. This led to heavy conductors and currents running throughout the vehicle. Modern vehicles use lightweight wiring to switch relays. If there is a relay it might have failed. find and remove the relay. If the light goes out substitute the relay with a known good one.

Check the reverse light switch. It bolts into the gearbox. Check the manual for location. Pull the plug off it, if the light goes out the switch needs replacing.

After that you will have to 'chase' the wiring; checking each wire back to source.