To adapt an automatic, you'd need a custom conversion. This would involve:
* Choosing a suitable transmission: This is the biggest challenge. You'd likely need a relatively small and simple automatic transmission from a much older car (possibly a very early Fordomatic or something similar from the 1950s or early '60s, but even those might require significant modification), or perhaps a modern aftermarket unit specifically designed for adaptation to older engines. The size and bellhousing pattern are crucial considerations.
* Fabricating or adapting a bellhousing: This is almost certainly necessary. You won't find a readily available bellhousing that will directly connect the flathead to any common automatic transmission. A custom bellhousing would need to be made or significantly modified.
* Custom driveshaft: A new driveshaft will be needed to connect the transmission to the differential.
* Torque converter: A torque converter appropriate for the engine's power output will be necessary.
* Hydraulic lines and controls: You'll need to adapt the hydraulic lines and controls for the automatic transmission.
In short, there's no "off-the-shelf" automatic transmission that will fit a flathead Ford engine. It's a significant undertaking requiring fabrication and specialized knowledge. It's more a project for a skilled mechanic or transmission specialist than a simple bolt-on upgrade.