There are i's to dot and t's to cross, but the idea is as follows:
I do not have your specific O-36 turnouts, but in hijacking the figure from the instruction manual I presume you have a situation as shown in the upper part of above diagram. You have 2 turnouts. You have 2 manual lever controllers. Each lever controller is essentially 2 electrical "momentary" switches with spring return to center OFF. Throw the lever one way and the turnout goes to Straight/Thru. Throw the lever the other way and the turnout goes to Diverge/Out. So you essentially have 4 total momentary action electrical switches.
The wireless relay modules mimics the behavior of the 4 momentary switches. There are 4 relays on the module. When you press button A on the fob, relay A on the relay module closes for as long as the A button is held down. This in effect is like throwing a lever on the manual controller in one direction. When you press button B on the fob, relay B closes for as long as the B button is held down. This is like throwing the lever on the manual controller in the other direction. Hence that's why I suggest a 4-channel relay module to handle your two turnouts.
Note that the relay module needs a source of DC power...typically 12V DC. Depending on what you are using for track voltage or accessory voltage there are inexpensive modules to convert that voltage to 12V DC. By inexpensive I mean a few dollars...less than $5 in any case.
And as an aside, I showed a 4 channel relay module by way of example. That would "exactly" accommodate 2 turnouts. But maybe you see your son's railroad "empire" expanding and you want to control additional turnouts or maybe turn an accessory on and off at will. There are all manners, combinations, permutations or what have you of relay modules with 1,2,4,8, etc. relay modules and corresponding fob hand-held remote controls.
Also, the Amazon prices I showed are not the best you can do if you are a frugal shopper. I am a frugal shopper! Those where essentially what popped up first. So when you say "least expensive" if you have DIY DNA you can get it done for not much coin! I realize there's a lot to be said for a single user-interface all under the Lionel umbrella. I get it. But I don't think it's the least expensive.
