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On my Z06 2004 with the manual transmission ( I know, they only come with manuals) there is a sensor on the side where you fill the oil. I was under the assumption it looked like the transmission drain plug. This sensor is probably on other years/models and if so, please post year/model of your Vette. This is not an all inclusive detail how to change the oil but info I did not see on other posts.
Tools needed.
3/8" Ratchet
3/8" X 3" extension
3/8" X 10MM Hex Head (Not TORX)
7/8" Combination Wrench
(I tried a deep well socket but not enough room)
Oil Pump
Capacity is right at 4 qts Dexron III ATF for the trans, and just under 2 qts of 75W-90 GL-5 Gear Oil, after adding the 4 oz bottle of GM Limited Slip additive for the diff. Fill to the top of the filler hole, with the car level.
If you have a small air compressor, I found that using air to push the fluid into the transmission works better than a pump.
Drill a hole in one of the fluid bottle caps to fit the fill hose and cut it to go all the way to the bottom of the bottle.
Drill a second hole to fit a small air line from the air compressor. Connect it to a trigger nozzle on the compressor, and set the air pressure low (10 psi or less).
Slowly pressure the top of the bottle, which forces the fluid up the tube and into the transmission. Takes less than 30 seconds to empty the quart bottle.
Mine started running out at about 3-1/2 quarts.