The engine only requires a "hot" side and a "cold" side, so there's no reason why the cold side can't be way below ambient temperature, such as an endothermic reaction or a tank of liquid nitrogen. Here's a video of a guy running one of those coffee-cup-top toy engines off liquid nitrogen: https://www.youtube.com/watch?v=O6eEvU5piQg

Some people have proposed this as yet another bulk energy storage system for the grid; giant underground cryotanks of liquid nitrogen. With careful design the same gear could be used in "forwards" or "reverse" mode. The limit is that the Carnot efficiency isn't great.

You could certainly run such an engine, very efficiently at that, from vaporisation of LNG.

A reverse stirling engine takes mechanical energy and produces a temperature differential. These kinf of setups are actually being used for quite efficient cooling.

It would still be a normal Stirling engine and not reverse. The engine only needs a large enough temperature differential. Where that comes from does not matter.