Neither specific websites of Hertz, Budget, Alamo, Enterprise, Thirftly or FOX allow you to specify that you want to rent a manual car. Neither do Kayak, Hotwire and Priceline.
Is there a way to book a manual car in the US?
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tl;dr: You cannot
Generally US rental companies don't have option to select transmission type, because they don't offer anything other than automatic.
The only cars in popular rental companies would have any chance to have manual transmission, would be high-end sports cars. And even so, that chance is slim to none. For example at Hertz you have "adrenaline collection" with cars like Mustang GT Premium or Corvette Stingray and they still only have automatic transmission option. They even have European sports cars like BMW M5 or Porsche Cayman, also only automatic.
The only way to get rental with manual, would be to use one of the peer-to-peer rentals and choose category "exotic cars" or similar. But even with these I haven't seen much offer of manual transmission (practically none at all).
Some companies rent manual cars. Here's one such company in San Diego.
I can't help but chuckle that they have a paragraph explaining the context, and have 'how to drive a proper car' (my wording) sessions.
I've not lived that long in the US, and finding a manual car to purchase was a struggle. We got a very fine two year old Saab, but gave up on the idea of getting a diesel powered vehicle.
In North America the percentage of drivers who can use a clutch and manual transmission without burning one of them out inside a week is rather small. Driving schools teach with automatic transmissions, not all that many drivers bother to learn about anything other than D and R.
The rental car companies now have a simple choice: buy cars everyone can use, or buy cars a lot of renters will either decline or damage. You also won't get diesel rental cars for the same reason - the engine will be ruined the first week because someone put regular gas in it.
I've rented in Europe (Germany) and got manual-transmission diesels every time. Also every time the rental agent heard my Canadian accent and asked me if I can drive a manual. I said I own one back home, which ended the discussion.