The problem is might be the reverse inhibit solenoid system which is a safety feature that prevents an inadvertent shifts into reverse at speeds above 3 mph. The system consists of the following components: 1) The engine control module (ECM) and 2) the reverse inhibit solenoid. With the ignition ON, battery voltage is supplied directly to the solenoid which unblocks the reverse gate. At forward speeds above 5 km/h (3 mph) the ECM opens the control circuit of the reverse inhibit solenoid. This de-energizes the solenoid and prevents the shift lever from going into reverse.
Battery voltage is supplied directly to the reverse inhibit solenoid actuator. The ECM controls the reverse inhibit solenoid actuator by grounding the control circuit via an internal switch called a low side driver. At driver has a fault line which monitors the control circuit. When the ECM commands the reverse lockout solenoid ON, at vehicle speeds below 3 km/h (2 mph), the voltage on the control circuit should be low, near 0 volts and reverse gear will be available. When the ECM commands the control circuit to the reverse lockout solenoid OFF, the voltage should be high, near battery voltage and reverse is locked out.
Is the check engine light on? If so, DTCs P0801, P088C, or P088D may be set. All of those are codes related to the reverse inhibit solenoid function. To solve those codes you need a scan tester and GM service data for the seven speed which is available by subscribing to ACDelco ESi.
The solenoid and its harness connector are not accessible with the transmission in the vehicle.
A faulty solenoid is possible and if the solenoid is faulty, transmission removal is required.
Unless you are an advanced DIY with a lift and the right tools, doing that yourself is really not an option. I'd seek help from your dealer or independent Corvette service facility.
