What to do when married to an alien who has no social security number and am filling as self employed?

If you file as married filing separately, use a fake SSN to pass the interview.  Try 999-88-9999.  Print your return and file by mail, you can't e-file.  White out the fake SSN first.  If your spouse is a non-resident alien, write "NRA" in the space for their tax number.  If they are a US resident but don't work, attach a written statement with their name, birthdate and address and a brief explanation.

If you want to file as married filing jointly, you must report your spouse's worldwide income, even if they have no  income in the US.  If they are living in the US and are not authorized to work, you must report their income anyway.  The IRS won't inform to ICE, they just want their money.  Again use a fake SSN, print your return, and white out the fake SSN.  Attach a form W-7 application for ITIN (international tax ID number) for your spouse.  Send the federal tax return, form W-7, and any required proof, to the address for the W-7 application.  After the IRS issues the ITIN, they will process the tax return.  When they mail you the ITIN, you can go back into Turbotax, replace the fake SSN with the new ITIN, and file your state tax return with the ITIN. 

If your spouse does not have a social security number or Individual Taxpayer Identification Number (ITIN) you cannot e-file the tax return. 

You will need to wait for your spouse to receive a social security number, if eligible, and enter it or follow the instructions above from @Opus 17 to print, sign and mail your return along with a form W7 requesting an ITIN for your spouse. 

There are other options available Here as well.

[edited 2/17/2021 4:16pm EST]