This worked for me.  I hope it works for you!  We know Intuit unilaterally vetos use of all zero SSNs and prohibits us from e-filing with that.  Furious?  Yes, me too!

I use QB Desktop Accountant 2022.  What does not work is trying to enter zeros by using the regular employee profile on the Employee List we generally use.  What did work was going at that employee profile through the deeper Payroll Setup.  On the Employee drop-down, click Payroll Setup.  From there click on Employees and get to the list of employees.  On the right side of the line select Edit.  On that screen enter your all-zero SSN.  Save, finish, exit out of Payroll Setup.

Once I did that, the e-file went through and the proper "Applied For" appeared on the W-2.

If someone gets this to work on other QB flavors, please post if the menu roadmap is different.