We live in Texas and the deadline was extended to June 15, but I need to file an extension for the October deadline. Can I do this through Turbotax?

Victims in FEMA Disaster Areas: Mail Your Request for an Extension of Time to File

To request an extension to file your federal taxes after May 17, 2021, print and mail Form 4868, Application for Automatic Extension of Time To File U.S. Individual Income Tax Return.  We can’t process extension requests filed electronically after May 17, 2021.

https://www.irs.gov/filing/where-to-file-paper-tax-returns-with-or-without-a-payment

No, you must fill out form 4868 on paper and mail it to the IRS. Electronic filing of the extension form ended for all taxpayers at midnight on May 17, 2021, the official deadline.  Taxpayers who have an extension past May 17 due to special circumstances, and want to extend to October, must file their extension request on paper, and the IRS will determine their eligibility based on their address on the paper form.