If you plan to file jointly, then it would be much easier if your wife has a Social Security or ITIN, so filing an extension would make sense. That said, you should be prepared to file without the SSN or ITIN if you have not received it by October 15th. An extension is only an extension of time to file the "return". It is not an extension of time to pay tax. Any tax due should be paid on or before the April 15 filing deadline.