My wife is a non resident alien, I will put NRA for her ss# number. But it seems like that means I will have to mail my taxes. Can I still e-file if I do this?

No, you will have to mail your tax returns.  You will need to input a dummy SSN (such as 999-88-9999) into TurboTax in order to finish your tax return.  After you print your return, make sure you "white out" the dummy SSN and write on that space 'NRA".  Then you will need to sign and mail  your return.