How would I enter locality name when it is just NY?

Your locality would not be NY as that would be the state.  Your locality is the town or city you live in or work in. 

Where are you trying to enter this?

If you are trying to enter this in the W2 section, then your locality would be whichever township or city  you live in.  Ex. Corning, Rochester, etc. If you are not in NYC, what you enter for your locality does not affect your return. on the W2.

Are you working on the state section?  If you are in the state section, and you do not live or work in Yonkers or NYC, then just continue through and say no, you did not live there or work there.  If you are in any other jurisdiction, you would need to file your local taxes separately through the local taxing authority as TurboTax does not cover local taxes in NY for places other than Yonkers and NYC.