Seat belt light doesnt come on when turn on ignition

No there is no light or chime

Ok, You will need to have the SRS control module scanned to establish if it is in communication with the rest of the vehicle, then diagnostics can begin. Since multiple sensors are not doing their job on the same circuit, the issue will likely be related to a control module.

The airbag light is working fine should there be a wire going to the seatbelt catch

The buckle portion needs at least one, but likely two wires running to it, but they should not be visible. We're missing a few things here. I need to know if the seatbelt light comes on with a passenger in the passenger seat and the seatbelt knuckled. Is the airbag on/off light functioning when the passenger moves in and out if the car? Is the airbag light and seatbelt light on the instrument panel working when you turn the key to ON without starting the engine? I have to have this information to help you.

No light with a passenger, seatbelt buckled or un buckled air bag light working correctly airbag light comes on with ignition in on position but not seatbelt light

Ok, look under the driver front seat, there is a wire that runs from the carpet to the seatbelt. Make sure it is not unplugged. Did you recently buy it and discover this, or did the problem just pop up?

Recently bought it and noticed no seatbelt light

Ok, There is also a procedure that someone may have done to shut off the seatbelt chime, but not sure if it shuts off the light as well. 1. Get in, shut the door, buckle the seatbelt 2. Turn key one click 3. Wait 10 seconds and start the engine 4. Unbuckle and buckle the seatbelt three times 5. Wait ten seconds and shut the car off 6. Start it and see if you get the chime and seatbelt warning