That's a pretty simple circuit. Power is fed to the chime module, then sent to the light at key switch ON. When the driver's seat belt is buckled, a switch in the buckle itself opens, breaking the circuit and turning off the light. Only the driver's buckle has the switch.
Try driving the car a short distance without buckling the belt. If you get a slow chime for the first few seconds after putting it in Drive, the circuit is okay and the seat belt switch is probably bad. Try spraying some contact cleaner in there first. If that doesn't work, I'd just pull the bulb. If there's no chime, it's likely the chime module, and I'd still pull the bulb.
Try driving the car a short distance without buckling the belt. If you get a slow chime for the first few seconds after putting it in Drive, the circuit is okay and the seat belt switch is probably bad. Try spraying some contact cleaner in there first. If that doesn't work, I'd just pull the bulb. If there's no chime, it's likely the chime module, and I'd still pull the bulb.