Turbo Tax said my taxes need to be filed via mail because my W-2 has a blank. However, the printout says New York requires taxes to be e-filed. Will my taxes be accepted?

New York requires the return to be e-filed unless it can't be.  In that case, they will still allow you to send it in by mail.  In this case, TurboTax does not support efiling for the diagnostic above, so you must mail it in.  Note their website:  https://www.tax.ny.gov/pit/efile/efile_mandate_for_individuals.htm says the following:

You're required to electronically file your return if you meet all three of the following conditions:

• you use software to prepare your own personal income tax return; and
• your software supports the electronic filing of your return; and
• you have broadband Internet access.

So when TurboTax cannot support e-filing, you can mail it in.