From the instructions for Form W-2 (and W-2c):
Correcting an employee's name and/or SSN only.
If you are correcting only an employee's name and/or SSN,
complete Form W-2c boxes d through i. Do not complete
boxes 1 through 20. Advise your employee to correct the
SSN and/or name on his or her original Form W-2.
That implies that a name-only error warrants a W-2c. The part about "advise the employee to correct the SSN and/or name on his or her original Form W-2" implies that you don't need to wait for the W-2c before you can file. Rather, you can correct the name error yourself on your original W-2 and use that when you file. As long as the SSN is correct, everything should match up.
- Request a W2-c from your employer.
- Correct your name on your original W-2.
- File with the corrected original if the W2-c doesn't arrive in time.