I'm filing from outside of the US (I live in Japan,) and my wife doesn't have a social security number because she's not an American. Can I not enter that field?

Yes, you can leave that blank and continue through the program. 

The program may show the missing number as an error, but should allow you to E-file.

 

You also have the option to elect to file a joint return. If that is your choice, TurboTax can help you with that filing as well. 

 

HERE are the options for filing when married to a non-resident spouse.