Thank you to everyone who answered. Her HR has said they will make up the difference on their side, meaning they will pay what should have been withheld to the IRS this year. I think the only further impact is that will be treated as income to us on our 2019 tax return. Assuming they handle as they have said, it seems it's been resolved and in a way that doesn't hurt us at all.
Responding the last comment from pk, I should have said marriage green card. Yes, I'm a US citizen. I'm not sure what five years of exempt status is, but she's been in this country less than 5 years as of 2018. I'm not totally sure if she was supposed to have FICA withheld in previous years or not. Maybe it's optional on a status like an OPT visa - checking her Social Security login it looks like her employer in 2015 actually did withhold FICA, but then the current one didn't in 2017 or 2016.
