No, you do not need to request a new W2 from your Employer because it has your married last name, just give them a copy of your new Social Security Card so they will have the correct last name going forward.
It can take the Social Security Administration 4-8 weeks to update name changes to the IRS. I would recommend that you first try using your maiden name to E-file your taxes. Be aware that if the IRS system has not been updated yet with your new information, your return will be rejected. If this happens, you will simply need to login to TurboTax, change your last name back to your married name and resubmit your return. This type of reject doesn't have any negative consequences on the filing of your Tax Return or the issuance of your refund.
