This page of the Weird West guide describes two ways of obtaining metal ore.
Weird West features various kinds of firearms (revolvers, shotguns, rifles) and melee weapons (knives, machetes). Weapons can be upgraded to increase their stats but you need metal ore to do so.
To mine metal ore in Weird West, you need a pickaxe - it can be purchased from shopkeepers in towns. The tool has limited durability and will break over time, so it's a good idea to have a spare one.
Exploring and trading
All crafting ingredients can also be found while exploring Weird West's world. Some merchants offer them as well.