Amateur hydroponicist here. It depends on what you mean by 'vertically'. You can certainly grow potatoes in containers that are stacked vertically. You can also grow them hydroponically. However, the issue I've noticed in the hydroponics community is that no one is interested in growing potatoes. That is really the problem with these vertical ag startups and such. They focus on ridiculous foods like greens, which -- while nutritious and easy to grow -- cannot form the bulk of a human diet.

As a community, vertical agriculture need to focus on high calorie crops like potatoes or sweet potatoes or at least something useful like beans.

But circling back to the beginning. You can't really grow a potato with less industrial input vertically than you can with regular land, so unless you are really out of land (and the United States at least is not running out of land anytime soon), it doesn't really make sense to do so. Potatoes are really easy to grow -- you stick them in the ground and dig them up a few months later. Anyone can just buy a few acres of land, fertilize it, stick in some seed potatoes, and get a pretty decent crop that more than covers their costs. This is currently way easier than the amount of setup it would take to use containers. If you were to use conventional growing containers, you would need to import large amounts of soil / substrate. If you were using hydroponics, you'd also have to buy large amounts of hydroponic substrate or expensive nozzles for aeroponics. Either way, it's more expensive.

Anyway... wish me luck, I'm starting some potato growing experiments this summer to see if I can develop new container, vertical, and hydroponic techniques. I'm particularly interested in growing potatos without a substrate and without expensive aeroponics. Currently investigating 'aeroponic' drip systems.