Supply depots are where you earn Supplies by killing Zombies with Weapons. Supply Depots are all based on "real world" businesses, located near you. You earn a new Supply Depot location for every 5 abandoned buildings you purchase.
Like Businesses, each Supply Depot has a Personality it's aligned with. Assigning a Survivor of a matching personality yields 20% more supplies per run. There does not seem to be a corollary between the survivor's level and the supplies yielded; a Level 0 Loner will yield as many supplies as a Level 9 Loner in a Loner-friendly supply depot.
As you complete supply runs, you earn access to better Weapons, and also different levels of supply run (Easy, Medium, Hard and Nightmare). The higher the level, the more supplies you can earn, but the more zombies attack you each time as well.