The stickys have a whole segment on PID controlled forges and ovens.
To convert that oven to propane, first you need to know what type burner you have. If it is a venturi burner, it has no blower and the air is drawn by the venturi of the gas jet. The air chokes affect the amount of draw and the combustion. The orifice size creates the draw. Natural gas has a larger orifice than propane does. The orifice will most likely need to be changed. You would want something around an .030-.035" " orifice. A photo of the burner would tell me if it is venturi. Before tearing it apart, try and determine the orifice size. If it isn't a lot off, the forge may run OK on propane with a little adjustment of the air chokes. It won't hurt to try it and see ( assuming it is venturi)
If it is a blown oven, you will have to add a proper size blower. Also you need an air valve and/or blower choke to regulate the air flow. If the entire manifold is missing, then you need to make that, too. Depending on what is still there, it may be easier to build a complete new burner. In a blown burner, the gas is directly mixed into the air flow in the manifold with no orifice. Any gas inlet pipe or tube size over 1/8" ( 1/4" is standard) will work. The gas flow is regulated by a needle valve.
In a simple PID control, a digital device called a PID reads the temperature in the forge and compares it to the temperature programmed. If the forge is lower , it opens the gas valve. If it is higher, it shuts it off. This requires a pilot flame for a safe operation. Control is a bit irregular because the flame is either full on or completely off.
In the two-stage PID I designed, the flame switches from a pre-set HIGH to a pre-set LOW instead of ON and OFF. No need for a pilot and very close temperature regulation. It costs less that $10 more to have this much more accurate control.