Can I use a new Mac mini as the operating system for my 2009 iMac?

No, if you get a new Mac mini you will need an external display, keyboard and mouse.

see > Connect a display to Mac mini - Apple Support

and > Set up your Mac mini - Apple Support

Then after you setup and migrate to the new Mac mini, you can recycle or re-purpose the old iMac.

see > What to do before you sell, give away, trade in, or recycle your Mac - Apple Support

and > Transfer to a new Mac with Migration Assistant - Apple Support

maddogkm24 wrote:

I have a 2009 iMac that no longer supports updates. Can I purchase a new Mac Mini and use that as the new operating system for my 2009 iMac?

That question does not make sense. A Mac mini is a computer – not an operating system.

If by any chance, you are asking about Target Display Mode, that probably won't work. Although some 2009 iMacs (24" and 27" models – but not 21.5" ones) support Target Display Mode using Mini DisplayPort input, Apple added restrictions such that no recent or current Macs support using any iMac as a Target Display.

See the bad news here: Use your iMac as a display with target display mode - Apple Support