No, there is not an API and due to the sensitive nature of a tax return it doubtful that there ever will be an API.
TurboTax users who complete and file a tax return using the online editions can always access the returns by signing onto their online account with the same User ID and password that was used to create the account.
TurboTax users who complete and file a tax return using the desktop CD/Downlead editions always have the return available on the computer where the TurboTax software was installed.