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No there is no such site.
You can try finding the build sheet on top of the gas tank but it probably will not be in very good condition if it even is still there.
Is there any place that you can put a 1968 Vette VIN number and get the build data for that car?
You can get a very close date of the build date with the vin number, probably the exact date if you do a little research.
The last 6 digits are the consecutive build number. If you post the number here I can give you the build date.
If you want to do it on your own, go to the C3registry and look up a vin near yours. Many have the trim tag pictured of there cars, if your vin is within 40 or so cars of another car it was built either the same day or one day before or after it.