If your spouse does not (yet) have a Social Security Number or an ITIN number, then the answer to your question is that you probably won't be able to
file an easy extension request electronically (through TurboTax or otherwise); but you can certainly do
it the old-fashioned way, on paper. Here's how:
Go to the
following IRS.gov website to obtain Form 4868 (a.k.a. Application for
Automatic Extension of Time To File U.S. Individual Income Tax Return):
https://www.irs.gov/pub/irs-pdf/f4868.pdf
Once you have that Adobe Acrobat document, simply print out Page
1. You can fill out the form itself by hand, and then mail it. In the
space that requests your spouse's Social Security Number, simply print
the words "ITIN TO BE REQUESTED." Please see also the instructions for
Part I (Identification), as found on Page 3 of the download.
It
does not matter if your spouse ultimately intends to apply for a Social
Security Number, or an ITIN number instead. You will use the same
wording on the extension request form, regardless of that distinction.
Thank you for asking this important question.