While I agree that significant currents are possible (depending on your definition of significant), the current will also be dependent upon the resistance of the load (typically the LED in this discussion). But the current is primarily dependent upon the dissimilar metals used in the thermocouple and the amount of heat. Power as you must know is the product of voltage times current.
In the most simple example, you will need the heat source, the appropriate thermocouple, an LED for the load and some sort of regulatory circuitry which will also consume power.
I go back to my previous statement that a battery will be cheaper for a few reasons. I will state a few here, but I will not try to educate the uninitiated, for I studied electrical engineering for my BSEE, for my graduate courses, and continued throughout my career from which I semiretired many years ago and cannot hope to pass on the entire course in a blog with my tired fingers.
Reasons or issues:
LED designed for a tight range of voltage. This must be supplied by the circuit. Too much voltage and the LED will be cooked quickly. Too little and it will just not conduct any current (no light.....dark.....no workee in either case!)
Heat source needs to be fairly predictable to be practical. Your pistol light will not work until you have fired it enough to generate that heat. You likely will need the light for your first shot. Oh yes, you could back it up with a battery (or some storage circuitry), but then you would be done....you would not need the expensive and complicated circuitry.
There are others, but I will not try to design this for you...the battery is still cheaper, more simple and will likely work as intented.
Yes it can be done. Is it practical? I don't think so. For many of the same reasons that powered flight can be accomplished with photovoltaic cells -- it can and has been done but due to the complexities (cost, weight, etc.) it is just not practical.
Light your fancy dandy gun light? Can be done but it will be expensive, heavy, costly and ugly as compared to a very small and simple battery which reliably supplies the voltage as the driving force at the current you need for a fairly predictable amount of delivered power to the LED load.
Just how much ugly, complexity, reliability and cost are you willing to sacrifice? This is a hobby proposition for someone with too much time and money on their hands.
And no, I will not draw the circuit for you, but define your heat source and conversion technology and it can be done.
