NYS lets taxpayers download other forms from their website (e.g. 1099-G), but I couldn't find anything for 1099-INT, and I haven't received one in the mail.
To be clear, I was due a refund for tax year 2019, but I didn't file my 2019 return till the summer of 2020 thanks to the COVID extension, and so the state kindly paid for the excess time they held on to the refund.
The interest they paid is above $10, so I would expect a form for it.