Yes, you can mail returns in NY state.
They highly encourage e-filing, but some people just do not have that option. It took a little digging but I did find the information of where to mail it in NY, here's the link: https://www.tax.ny.gov/pit/file/return_assembly_mail.htm.
The IRS was a bit easier to find: https://www.irs.gov/uac/new-york-where-to-file-addresses-for-taxpayers-and-tax-professionals.
