It's a liquidity problem. Retail absolutely can drop any given bank into a liquidity crunch by pulling out too many funds, too quickly. It doesn't even need to put a given bank at risk of insolvency, if the situation is read as widespread and/or growing, because as the event expands, so does the likelihood that someone else is mismanaging their books. Someone who is hooked into another institution, and another, and another. Contagion.

Anyway, corporate depositors have a duty to safeguard their capital. That means that if a bank run is underway by retail depositors, they're in line too, willing participants or not. This is why, again, even discussion of bank runs is discouraged, and their likelihood and effectiveness downplayed. They're built on turning the imperative of self-interest, which the financial industry is built on, on its head.