Does Turbotax online Form 4868 support filing an extension for a gift tax return (Form 709) as well as individual taxes (Form 1040)? If yes, is this a free service?

TurboTax can produce the Form 4868 for you as part of your tax filing process.  Form 4868 is provided, no cost, in all versions of online and all versions of Desktop.

There are two ways to request an extension for Form 709, United States Gift (and Generation-Skipping Transfer) Tax Return, and the first one is as you suggested:

1. Any extension of time granted for filing your individual federal income tax return will also automatically extend the time to file your federal gift tax return.
   • Use Form 4868, Application for Automatic Extension of Time to File U.S. Individual Income Tax return, or Form 2350, Application for Extension of Time to File U.S. Income Tax Return for those living outside the U.S..
   • Use these forms to extend filing time for a gift tax return only if you are also requesting an extension of time to file your individual income tax return.
     
     
2. If you do not request an extension of time to file your individual income tax return, file Form 8892, Application for Automatic Extension of Time to File Form 709 and/or Payment of Gift/Generation-Skipping Transfer Tax, to request an automatic 6-month extension of the time to file.

• The 8892 also serves as a payment voucher for a balance due on federal gift taxes for which you are extending the time to file.   Form 88992 is not supported by TurboTax.  Instead, you can find that form here
  https://www.irs.gov/pub/irs-pdf/f8892.pdf   It is a fill-in the blanks form that you can print and mail.
  https://www.irs.gov/forms-pubs/about-form-8892