My wife is here on an O2 visa and she was explicitly told not to get a SSN. I would like to e-file my tax return without a SSN for her

[LinaJ2018]

Per IRS, you cannot e- file a tax return which does not include the social security number.  You will have to apply for an Individual Taxpayer Identification Number ITIN form W-7 for her.  Please see attached.

https://www.irs.gov/Individuals/General-ITIN-Information

You will attach either form 1040 or 1040NR ( Married Filing Jointly or Married Filing Separately ) to the W-7 with the required documentation and mail them together to the IRS address at Austin TX.   You can still use TurboTax to prepare the form 1040, but just no e file.  If your spouse is filing form 1040 NR, you are suggested to use Sprintax to complete the NR form as TurboTax does not support it. 

However, you might be able to file Single and do not need her identification number if you meet the requirements of Married Person Who Lived Apart.