Can my wife use my social security number to file her taxes even though we are separated but not legally separated without my permission?

A joint tax return as Married Filing Jointly cannot be filed without your permission.

If filing as Married Filing Separately then the name and Social Security number of the spouse is required by the tax code to be entered on the tax return.

Can you detail how she is trying to use it? She has to enter yours and you hers, if you're filing married but separate returns. The federal return will require it and any state returns will likely require the spouse's SSN also. 

You may find some additional help here:

https://www.justice.gov/opcl/overview-privacy-act-1974-2020-edition/ssn 

for married filing separately she must have your SSN and yo must have hers and they must be included on the other spouse's return.  The IRS uses the SSNs to check to see that both taxpayers either use d the standard deduction or itemized deductions  - no double dipping. failure to follow the rules will result in getting iRS notices and possibly tax bills. 

Let me answer differently.

You can't file jointly unless you agree.  That seems unlikely here.  So you both need to file as married filing separately.

If spouse A does not include spouse B's SSN, spouse A can't e-file.  They have to file by mail and attach a written statement explaining that they do not have their spouse's SSN, and they give the IRS as much identifying information about their spouse as they can (name, birthdate, last known address, etc.).  This will delay processing and payment of any refund, and could bring additional scrutiny on both spouses.

This applies to you as well, if you file MFS, you need your spouse's SSN or you must file by mail and attach a written explanation.