I bought online version in 2016. I need to file an amendment now, but the website only allows me to download a windows version. However, I don't have a windows machine.

Do you see the option to amend using TurboTax online when you click on the Amend(change) return link?

See this TurboTax support FAQ for amending a 2016 tax return originally completed using the online editions - <a rel="nofollow" target="_blank" href="https://ttlc.intuit.com/questions/3402261-how-do-i-amend-my-2016-turbotax-online-return">https://ttlc.intuit.com/questions/3402261-how-do-i-amend-my-2016-turbotax-online-return</a>

Or do you mean a 2015 return you did in 2016?  Do you have a Mac?  I see you are posting from Linux.

Actually, it's for my 2015 tax.

The website says:
 How to amend your 2015 return
1.Download your .tax file
2.Then download TurboTax Desktop 2015
3.Finally, open your .tax file using TurboTax Desktop 2015, and amend your return. Learn more

I have a mac machine.

Use the link below to request your 2015 MAC amend software 

https://ttlc.intuit.com/questions/3449967

Thanks! Is there a way to expedite the request? It says "Requests will be completed within 14 calendar days." But I need to mail my amendment in two days.

It's usually right away. Watch your emails.

You will want to go ahead and submit the request. Most requests are processed within 48 hours.

I received an email, and filled in the code at <a rel="nofollow" target="_blank" href="https://ttlc.intuit.com/questions/3978848-amend-prior-year-return-mac-software-request-download">https://ttlc.intuit.com/questions/3978848-amend-prior-year-return-mac-software-request-download</a>.
After I click submit, it shows Loading..., and then stuck there.

I really need this software now, I don't have much days before the IRS deadline.

Contact TurboTax support during business hours for assistance with the Mac download.
See this TurboTax support FAQ on how to contact Support - <a rel="nofollow" target="_blank" href="https://ttlc.intuit.com/questions/3401489">https://ttlc.intuit.com/questions/3401489</a>

That basically gives me three options:
1. Posting your question to our community of experts; or
2. Talking one-on-one with a live TurboTax Expert, CPA, or enrolled agent (requires the PLUS add-on if using Free Edition).
3. If you're using the TurboTax CD/Download software or can't sign into your TurboTax account, click here to contact us. Ask your question, click Continue, and follow our onscreen instructions.

I believe 1. is what I am doing here. 2. requires me pay another $30, 3. directs me back to 1.

What exactly are you suggesting me to do?

Try using the contact website and indicate that your product is a CD/Download(Mac) and then choose the Deluxe edition.  Then enter your problem about the Mac download - <a rel="nofollow" target="_blank" href="https://support.turbotax.intuit.com/contact/">https://support.turbotax.intuit.com/contact/</a>

Or you can use these websites to contact support -
Support can also be reached by messaging them on these pages <a rel="nofollow" target="_blank" href="https://www.facebook.com/turbotax/">https://www.facebook.com/turbotax/</a> and <a rel="nofollow" target="_blank" href="https://twitter.com/TeamTurboTax">https://twitter.com/TeamTurboTax</a>

What IRS deadline?   Unless you are amending because you received an IRS letter that told you to amend by a certain date, you have 3 years from the due date of the return to amend for a refund.   For a 2015 return, that is 3 years from April 18, 2016.   Other than that there is no current deadline for a 2015 amendment.  

If you owe then amend ASAP to avoid additional late charges, but a few days will not make much of a difference.

OP is asking from Linux.

OP said he had a mac