That cart is pre 1970 which at over 40 years old, it does have a certain cool factor to it.
I believe that the carts back then were 24v so you can use any type of control device from forklifts, sweepers, manlifts to golf carts. They are probably a series cart that used some type of stepping resistor to vary the speed.
For simplicity, I would find an old junked cart and take all the electrical/electronics from it. If it were me I would look for an old EZGO between 1996 and 2001 that has a series set up with a controller. With that you would have a throttle, controller, F-N-R switch, micro switches and solenoid. You may want to go to 36v with this cart, the motor should handle it and then all the controls would work as well. If you stay with 24v, make sure the controller and solenoid will work for you or you will be buying those after the fact.
Wayne Manufacturing bought Autoette which made Golfmobile in December, 1954 and sold in 1970. Wayne Manufacturing was bought by FMC in 1972.
Here's some history for you:
The Autoette was first manufactured in early 1948 in Long Beach, California (USA) by Royce Seevers. The company was then sold to Blood Sales Co for a short time before being resold to Wayne Manufacturing Co; a street sweeping company who moved to Pomona, California. After some legal problems, the Autoette was then sold back to the Seevers' family, lock stock and barrel in 1958. From then until 1970, they controlled manufacturing, distribution and sales of Autoette vehicles.
The Autoette vehicles were electric cars that came in a number of forms that range from pickup trucks (that could carry up to 1/4 of a ton), cars, and golf mobiles. Ranging from 2 - 4 seats, the Autoette had tiller steering and was powered by an electric motor which drew its power from large 24 volt batteries. Production ceased in the 1970s.