World of Thedas page 35
"Orzammar relies upon the surface not just for its prosperity. but for its survival. Ages of Bligths have taken thousands of thaigs away from the Dwarves. These were the places where most of the food was raised. The dwarven kingdom that endured alone, independent beneath the Stone from time immemorial, perished in the First Blight, faded into myth. Now the remaining dwarves underground cling to existence through a lifeline to the surface, a chain forged from the casteless."
Orzammar would literally perish if it managed to anger humanity as a whole. They don't need to invade, they just need to stop selling them food.
Which is kinda funny given that large-scale long-range food transport in quantities enough to be relevant is virtually impossible in this period without riverine or ocean transport. Land-based shipments of food would be so fantastically expensive and low-volume that it's hard to imagine Orzammar managing it somehow.
Or, for comparison: historically, the Roman Empire sustained the outsize population of its capital (Rome, then later Constantinople) via state-subsidized shipments of grain from Africa. It was one of the largest items in the imperial exchequer, alongside the army itself. (In related news, water transport of foodstuffs was even needed to sustain Rome's field armies, which were therefore mostly located on rivers, like the Rhine and Danube, or near large bodies of water, as in coastal Syria. Each army was akin to a city, especially when concentrated for campaigns; they had to move almost constantly or risk running out of local forage, because their wagon trains couldn't sustain the entire army with food.) So the Roman Empire spent fantastic sums of money feeding Rome with African grain; when the African grain route was cut (440s for Rome, 610s-40s for Constantinople) the cities starved.
Orzammar isn't Rome, and it doesn't have the sea route to reduce costs to something approaching a manageable level. Unless there's some other water route into the Deep Roads - or unless they can magic the food down there somehow - it's fantastically implausible that Orzammar would import enough of its food from the surface to matter (let alone enough for the dwarves to be
reliant on that supply).
Or they might have figured out railroads already. Either way.