Was the work she did of the nature that the S Corp is formed for, or does she have a "side gig" as well as the S Corp? Let's review that a 1099-NEC is informational, and for an S Corp, it isn't required. That doesn't make it an error. But if that has her SSN and not the EIN, that makes it an error if these customers hired the S Corp, and she did the work.

"and no W-2s have been issued—just 1099s"

From whom?

That's the mix up: Who the customers hired and paid vs who did the work. The more common mistake is someone uses the S Corp for the customer relationship, then doesn't go through payroll for the person doing the work, and then the S Corp pays and issues the 1099-NEC to the person doing the work, trying to bypass the corporation having to do payroll.

"What are the potential penalties or implications of not running payroll through the S Corp and instead reporting compensation on Schedule C?"

Why did an S Corp get formed? Is she really using the S Corp, or is it a shell and never followed through? It's not just payroll. The contracts for services should be in the name of the corporation; she signs business paperwork as President in the name of the corporation. She is supposed to get insurance, worker comp, and execute other business agreements in the name of the corporation.

Did she ever/never do any of this? Did she ever/never issue W9 to clients in the name of the S Corp, so they knew they are not hiring her, personally? Did the S Corp ever file as an employer for the Feds and State? Was there an 1120S? Or was everything always on the Schedule C? Or, was it just her "income" on a Schedule C?

Perhaps there is no error.