If I claim my adult son on our 2019 tax return, would we be required to pay a penalty if he did not have insurance in 2018 and did not apply for a waiver?

[Vanessa]

Your 2019 return filed next year, there will be no penalties. The new tax law removed them for 2019.

Your 2018 taxes that we are filing right now, yes, there would be a penalty if you claimed him and he didn't qualify for an exemption.  Also, if you are claiming him, your income would be counted when applying for an exemption, so if he was claiming it was not affordable it would be based on your income not just his.