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Your Corvette have a year? It would help if we knew which one. Which relay is clicking? Continuous clicking, ign off, you have a badly discharged battery. No cranking can be due to a lot of reasons, poor battery cable connection, defective battery, or VATS. Measure the battery terminal voltage while someone holds, "crank", position on the ign sw. Battery voltage must not fall below 9.0 volts (during cranking). Below 9v, bad battery cable connection or battery discharged or defective. What is the battery voltage no load? 12.0 volts and below, discharged, charge the battery up with a wall plug battery charger.
If the relay click is coming from the start enable relay, then you likely have a defective clutch safety switch (gear selector sw at base of gear shift if automatic). Jump the safety sw and try a crank. There should be 12v on the jumper when the ign is in crank. If you jump 12v to the jumper, the starter should crank. Next you need to check if the VATS module is measuring the key pellet resistance. Remove the kick panel above the drivers feet and find the 2 wires from above the steering column that goes to a 2 pin connector and unplug. Insert key in ign and measure the resistance across the 2 wires from the steering column and compare this resistance to the pellet. Resistance cannot be more than 4% difference or you need a new ign lock which will have new contacts that make connection to the pellet.