When i turn on my range vent the vent leaks water done. Only happens mostly when outdoor temp is 0 or below
my WAG: there is some amount of house air (warm and moist) exiting through the vent continuously. As soon as it hits freezing cold metal, there is condensation and freezing. When you turn on the fan, the abundance of heat melts this ice (which can be quite thick) and it runs back inside the house and drips out where you can see it, and probably also where you cannot. I'd worry that the nuisance on your cooktop could be a worse problem inside the walls, dampening the insulation.
What you can control is how well insulated the duct is; you may be able to foam it in or something so the dewpoint is reached outside the walls. Perhaps cleaning, re-sealing, or replacing the outside damper so it closes better. The moving air is carrying the moisture; stop the air movement completely and there will be no problem in the walls. But a 100% sealed damper is an elusive quarry.