If you are a US citizen or US resident and your spouse does not have a Social Security number or an ITIN and you are not applying for an ITIN with the tax return then you can only file your tax return as Married Filing Separately. Where asked to enter the spouse's Social Security number enter 999-88-9999. You can only print and mail your tax return, it cannot be e-filed. When you print the tax return erase the Social Security number for your spouse and manually enter NRA for non-resident alien.
See this TurboTax support FAQ for the procedure to print and mail a tax return using the online editions - https://ttlc.intuit.com/questions/1944348-how-do-i-print-and-mail-my-return-in-turbotax-online
If you are a US citizen or US resident and your spouse does not have a Social Security number or an ITIN and you are not applying for an ITIN with the tax return then you can only file your tax return as Married Filing Separately. Where asked to enter the spouse's Social Security number enter 999-88-9999. You can only print and mail your tax return, it cannot be e-filed. When you print the tax return erase the Social Security number for your spouse and manually enter NRA for non-resident alien.
See this TurboTax support FAQ for the procedure to print and mail a tax return using the online editions - https://ttlc.intuit.com/questions/1944348-how-do-i-print-and-mail-my-return-in-turbotax-online
In general, the IRS requires the spouse's SSN so they can compare the two spouse's tax returns, because when spouses choose to file MFS, there are some credits and deductions that are reduced or not allowed and some other special rules apply. In your case, you need to indicate "NRA" in the space for the SSN, and that requires printing and mailing the return. Turbotax does not support e-filing for an NRA spouse.
It's a pity that Turbotax doesn't support e-filing in this case.
A spouse who is Married Filing Separately is not required to provide the Social Security card for the other spouse, although the return cannot be e-filed without the spouse's Social Security number.
I believe the IRS e-filing schema has been updated to permit e-filing when filing MFS with an NRA spouse. However, Turbotax has not included this feature.