Jim, first of all, welcome to the World of Windups!
The following is just my personal observations, so take it for what it is worth: There is an old saying that Lionel made trains by the thousands, while Marx made them by the tens of thousands. Marx produced windups from 1935 (or earlier, if you want to include Joy-Line) all the way up to 1975. During those four decades, the Marx windup sets were among the lowest priced way for parents on a budget to get a train set for their children at Christmas. It is still amazing to me how many Marx windup trains are out there. Now, contrast that to Lionel's mechanical train production... which I believe ran for a few years from 1931 to 1937 (I'm far from being an expert on Lionel, so I will defer to others if I'm mistaken). There were very few years of production compared to Marx, or even Hafner, Ives, and Flyer. Plus, those years started in the depression, which I would assume limited the market for toy trains in general. So, I think it's safe to say that the scarcity of Lionel compared to other domestic mechanical train manufacturers are why they are rarely seen in the posts.
The scarcity of Lionel mechanical trains also mean that I've had very few come across my workbench. I don't have nearly the experience with the Lionel offerings as I do the other domestic windup trains. So, take this observation with a grain of salt - it seems to me that Lionel windup motors tend to be more fragile than most of the other makes. Now, it's a bit strange because the motors are actually well built for the most part, but the ones I've tinkered on seem to have some Achilles heel that renders them inoperable. I don't know how many of the little Lionel Commodore shells I've seen over the years without a motor... and when I finally got my hands on one with a motor in it, the motor had whistle flap lever wadded up in it. The Torpedos seem to have gear failure on the whistle drive, and since the whistle functions as the governor, they can't be run without it. I even added a governor to one (in place of the whistle) for one of my friends on the forum... and it worked for a while until the ratchet decided to turn itself into scrap metal. I still have to tackle that repair someday...
So, you put those factors together, and it makes a running Lionel mechanical a pretty rare item, at least when compared to the other four common US makes.
But, getting back to your question, there may be some Lionel collectors out there than can help you with an original motor for your 1816... I'm afraid I can't begin to point you in the right direction to find one. I would suggest that you consider - as a temporary step - installing a Marx windup motor in it's place. They are common and cheap, so no harm in modifying one to fit the Lionel shell. You will need to fabricate some sort of custom mount, but I would think that could be done without making modifications to the shell itself. Even if you find the correct Lionel motor, you could run it with the Marx motor so you don't have to risk damage to the original Lionel motor.
I have a few Lionel windups in my collection, and they are pretty much shelf queens. I just don't want to take the chance of having a motor problem, especially when I have so many common windups I can run instead. I do have a Torpedo that I bought as a shell without a motor, and I installed a Marx motor in it so I can run it on the windup layout.
One final thought, if you don't have this book already, I highly recommend "Lionel Mechanical Trains, 1931 to 1937" by Richard T. Claus. It's an excellent reference for Lionel mechanicals.
I hope that you find a motor for your train, and I certainly welcome seeing more posts about Lionel mechanical trains!