Is it legal for an employer to claim that they pay income taxes on employee's behalf and therefore write the paycheck for the net amount owed to the employee?

Legally, she needs to provide the W-2. If she does not, you should contact the IRS and complain that your employer has not provided a W-2 to you; they may then contact the employer.

However, the important thing to note is, your obligation is not to have a W-2--it is to pay the taxes you owe. If you know your gross salary or wages (i.e. the total) and you also know your net, you can take that to an accountant--or use Turbo Tax or the equivalent yourself--and calculate what your taxes should be. The working assumption would be that the employer properly paid withholding. You would then pay or request a refund, as appropriate. If it later turns out the employer did not properly remit withholding on your behalf, there would be a legal issue to work out between you, the employer, and the IRS, but you won't know that until/unless it becomes an issue. Even if they provided a W-2, that would not guarantee the money had been properly remitted--a W-2 is just paperwork, and paperwork can be incorrect or false. So carry on as if the difference between the net and gross was withheld the way it should have been and honor your obligations on that basis.