Assuming (big assumption) that someone there discovered the relationship between pressure and temperature, they would have had the manufacturing technology to make a cryogenic refrigeration unit. (They're really not that complicated. Would be expensive, but a competent blacksmith could do it.)
Of course, then you have to power the thing. No way you could do it feasibly to make any significant quantity with just muscle power. But if they've worked out the relationship between pressure and temperature well enough to build the cooling unit, you just run it in reverse and you have a stirling engine.
Of course, now that you have stirling engines, you're rapidly leaving the mediaeval technology level behind...
And then there's the fact that cryogenic weapons just won't be that effective compared to thermobaric weapons. The maximum temperature differential you can cause by cooling is probably about 200C. You can easily cause 1500C worth of temperature differential by heating, probably more. The higher the temperature differential, the faster the weapon will be effective.
So, yeah, you can suffocate people with liquid nitrogen, but you need either an enclosed space, or really quite a lot of it. Freezing things requires extended contact with it, and a simple raincoat will provide a great deal of protection. A grenade worth probably wouldn't be that useful. You can find videos of physics instructors playing with that much with no side effects.
Much more likely would be to use the liquid oxygen acquired by liquefying air as an oxidizer for charcoal. It's a lot more potent than saltpetre, and gets you an explosive more powerful than TNT. If you want, you can use the liquid nitrogen to convert more of the heat into pressure as well. Depends on if your goal is to light things on fire or crush them.