I am basing my answer on the excellent Last Call by Daniel Okrent. It is a fairly good history of the entire Prohibition movement up through repeal of the 18^(th) amendment. While the more famous part of Prohibition was the amendment itself, the actual means to enforcing it was the Volstead Act of 1919. This act laid out several exemptions:

• Any alcohol owned prior to January 16^(th), 1920 was exempt (grandfathered in/no ex post facto laws)
• The "fruit juice clause" -- in order to allow farmers to "conserve their fruit" every head of household was allowed to ferment up to two hundred gallons of fruit juice. This led of course to a thriving industry of ready-made fermenting kits (which Okrent calls the California Grape Rush of the 1920s). To give an idea of the size of this, in 1917, 70 million gallons of wine was drunk but in 1925 150 million gallons of homemade wine was drunk.
• Sacramental wine.
• Medicinal alcohol. (A doctor's ability to prescribe it, however, was regulated. So a doctor could prescribe a dram of whiskey but not a series of cocktails. Sadly.)
• Anything containing less than one half of one percent alcohol was not considered intoxicating and not covered by the Volstead Act, depending on the product. Homemade ciders or wines was not covered by this but beer was. (For example, root beer contains a small amount of alcohol depending on how it is made; homemade cider could exceed that limit.)

These are, as best I can find, all of the relevant exceptions; I tried searching across multiple other sources to see if there were any exemptions for cooking wine but found no solid leads on that. (I also do not know if anyone attempted to argue the point that the act of cooking wine in food removed its alcohol thereby getting around the prohibition on sale of alcohol. This still leaves the question of how the restaurant even got the wine in the first place, of course.) If you were a chef at a restaurant and you wanted to make a proper coq au vin you were fresh out of luck as best as I could tell. If you were cooking at home and had laid down a large supply of wine before Prohibition then you could certainly use it that way, and I imagine some people did, though I would imagine you would prefer to drink most of it.

Edit: After a lot of searching and wrestling with Google cache I did find a short blurb from the Sun Sentinel that claims that salted cooking wine was allowed, though no sources are listed: https://web.archive.org/web/20210629050040/https://www.sun-sentinel.com/news/fl-xpm-1998-03-19-9803160145-story.html They also claim that many French restaurants closed in response to Prohibition -- I'm going to take a look into that claim. That may prove fruitful.

Edit 2: I believe this is my fourth or fifth answer in AH about booze. I remember writing about George Washington's production of applejack and at least one or two other things about Prohibition. Is there a way to find posts I made in a specific subreddit? Reddit search is terrible.

Edit 3: The full text of the proceedings of the 66^(th) Congress is at https://tile.loc.gov/storage-services/service/ll/llsl//llsl-c66/llsl-c66.pdf (warning, 250mb PDF). I am going to double check to see if I can find any exceptions for culinary purposes.

Edit 4: I've read H.R. 6810 through once or twice and I don't see exceptions for culinary purposes. It does in places mention nonbeverage uses but only in relation to sanitizers or industrial alcohol as best as I can tell.