Your surname does not automatically change upon marriage. If you officially change your name when you marry you must update your name with government agencies -- such as contacting the DMV in order to correct the name on your driver's license.
If you provide documentation of your name change to an employer, it is possible that an employer would allow you to work, but more realistically an employer is going to want a new employee's name on her ID to match the name on her work permit. Similar issues may arise with government agencies.
The woman should thus update her name with the USCIS.

