Too late to amend my 2015 return online, but I can't find where or how to download the desktop version and .tax file to amend it.

The amend software should be available on your 2015 Timeline after the 2016 online editions are released sometime during the first week of December.

Or -

If you paid for one of the 2015 online editions of Plus, Deluxe, Premier or Home and Business, then you will need to contact TurboTax support and speak with an agent to get a 2015 desktop installation file pushed to your download account.

Download the tax data file from your 2015 online Timeline.  See screenshot.  Then open the file with the 2015 desktop program.

• Click on Federal Taxes
• Click on Other Tax Situations
• Under Other Tax Forms
• On Amend a Return, click on the start or update button

See this TurboTax support FAQ for a contact link and hours of operation - https://ttlc.intuit.com/questions/1899263-what-is-the-turbotax-phone-number

To contact TurboTax customer service/support use their contact website during business hours.  

Submit your question/problem during business hours and then use the option for Call (a phone number will be provided for Call) to speak directly with a TurboTax agent -

https://support.turbotax.intuit.com/contact/ 

Note:  The Call option will not be available when TurboTax Support is closed.