The main New York tax form (IT-201) is not expected to be available for e-filing until tomorrow, January 13. That's why it's not giving you the option to e-file. Some other New York forms will not be available until January 27. Those dates are subject to change, and they don't necessarily mean first thing in the morning. Try again after tomorrow, or wait until IRS e-filing starts on Jan. 24. See the expected availability dates at the link below.
TurboTax will let you "e-file" your tax return now, but it will not really be filed. TurboTax will just hold it until the IRS begins accepting returns on Jan. 24. Nothing will be done with it before then. If you submit it early, be aware that you can't cancel it or get it back to make changes or corrections if you find that you made an error or left something out.
State forms availability table for TurboTax individual (personal) tax products
