How can I find the purchase date of my iMac?

I would try

https://checkcoverage.apple.com/

The purchase date is not the date you need. What you need is the year designation of your Mac that's in the About This Mac window:

That's the date designation that will determine what you can upgrade the system to.

I do not know my purchase date for my iMac 21.5 mid 2011 computer so I cannot access any upgrade downloads