Please note that when your payer issues an income document, they also report it to IRS/State Tax Office. In case they have provided you with a form without a taxpayer ID or incorrect taxpayer ID, it's recommended to request a new copy.
Your employer should be able to issue a corrected form with the only change being the Social Security Number.
In case this can be done before the tax filing deadline, please wait for the W2-C to be issued, however if due to any reason this cannot be done by the tax filing deadline and you owe money to the tax office, please file using your current copy and still request a corrected version from your payer. The IRS may send you a notice to confirm your taxpayer ID to which you'll have to respond to with the requested documents.