My wife's current Social Security number is not updated with her new name. Can I still jointly file her Federal and State taxes or will she have to do them herself?

You can still file with your wife, but in order to file the tax return, the IRS uses the Social Security Administration database to match the social security number with the last name.  You definitely want to update the name with the Social Security Administration but either update and wait to file OR file now with the old name then update.