Hello. The vacuum circuit for your headlights begins at the rear of your engine's intake manifold in between the distributor and carburetor at a fitting slightly offset to the passenger side of the motor. There should be a hose leading away from this fitting with a small in line filter just a few inches up from the fitting on the intake. And a few more inches up from that around usually close to the firewall is a vacuum check valve with 2 outlets on it. This check valve fails regularly and will let the headlights come up at wide open throttle when it does because it lets the vacuum bleed out of the reservoir. Which brings us to the next component. The big hose leading away from the check valve goes to the vacuum reservoir.
There are a few styles of vacuum storage canisters used over the years for the C3 with the best solution being the last. At the end of the C3 the reservoir was a thick tube that ran across the nose of the car like a bumper but down low. This tube was solid, did not flex under vacuum and lasted the long haul. In the beginning the C3 used a large canister that mounted next to the master cylinder under the LH fender. Then it moved to the nose of the car and took the shape of a coffee can. Both of these early styles were made with thin walls and would flex slightly under the changing vacuum conditions and it would fatigue the metal and make the reservoirs fail. So it is always a good idea to cap off the reservoir, draw it down into a vacuum, and make sure it holds vacuum. If not find a good quality aftermarket replacement with thicker walls that will not flex.
From the reservoir the vacuum leads to the headlight door vacuum relays. These are in the nose of the car just under the v tip of the hood. These vacuum relays fail very regularly and if you haven't replaced them yourself and in the last few years they are likely bad. gross vacuum leaks can easily be heard from the bottom of these vacuum relays. And the vacuum relays send that vacuum to one of two sides of the headlight door vacuum actuators. These are also in the nose of the Corvette next to the relays. They are large round canisters with a vacuum hose on either side. These can be tested by removing both hoses applying vacuum to one side and making sure it actuates and that there is no vacuum present on the pot on the other side. The actuators do not fail too regularly but if they are very old then they might well be in need of replacing. They are just rubber diaphragms inside. Thin rubber diaphragms.
The small hose leading away from the check valve at the rear of the intake manifold goes inside the car. It passes through the headlight switch for automatic operation and it also passes through an override switch below the steering column. From there it goes back out through the firewall and to the nose of the car and it plugs into the tops of the vacuum relays. This smaller diameter hose is the control side of the circuit or the primary whereas the larger hose with the reservoir that drives the actuators would be the secondary or the load side of the circuit. Wilcox has some really cool diagrams out that can help to better illustrate this circuit. Here..