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?
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).