Is my wife allowed to use my social security number to file her taxes without my permission we are separated but not legally?

Yes, 

If your wife is filing Married Filing Separately, she should list your Social Security Number on her separate return. 

You will should to do the same, list her social security number on your separate return. 

 

You also both need to use the Standard Deduction or use Itemized Deductions, one cannot Itemize and one take the Standard Deduction. 

 

According to the IRS:

“You must enter your SSN on your return. If you are married, enter the SSNs for both you and your spouse, whether you file jointly or separately.”

 

HERE is more information from TurboTax