Yes, you can go ahead and file your tax return.
When you enter your W-2 in TurboTax, you don't need to enter your SSN. In the W-2 section, choose 'I'll type it in myself'.
Enter your Employer Identification Number, their name, address, etc.
TurboTax automatically fills in your correct SSN that you entered in the MY INFO section.
Just save the corrected W-2 for your records when you receive it.
Here's more info on How to Manually Enter a W-2.