"but that's not relevant to my question"

You asked about Filing Requirements for an S Corp. That would include payroll, and if that had not been done, you would not move on to income tax filings or business filings.

So, yes, it's very relevant.

Did you use the googles:

https://nystax.custhelp.com/app/answers/detail/a_id/4184/~/what-is-required-to-file-as-a-new-york-st...

https://www.tax.ny.gov/bus/ct/ctidx.htm

https://www.tax.ny.gov/forms/corp_s_forms.htm

https://www.tax.ny.gov/bus/ct/s_corporations_post.htm