Turbo Tax does not allow extension beyond May 17. Do I need to go to the IRS website and file Form 4868 electronically to get an extension for the Texas June 15 deadline?

too late. the extension was due 5/17/2021.   if you don't owe there will be no penalties for filing late. if you need to file a state return that could be a different story. 

the June 15 filing deadline is automatic for those who qualify based on your state of residence.

It is not an extension.

You cannot request it.