Can I receive a direct deposit on an amended tax return when e-filing with TurboTax? A mailed refund is the only option I am currently shown.

This is an option available for this year per the IRS link below:

• IRS Amended Returns 

"Is direct deposit available for electronically filed Form 1040-X?
Beginning in processing year 2023, Direct Deposit can be requested for electronically filed amended returns for tax year 2021 and tax year 2022. The bank account information should be entered on the electronically filed Form 1040-X (or Corrected Form 1040-SS/PR)."

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You can, but only if the original return had direct deposit specified.   If the original return did not have direct deposit (and I filed a paper return), then I could not set it up for the amended return. Otherwise, it should work.