Do I need to upgrade my machine in order to release app updates on the App store?
macOS High Sierra 10.13.6 is as far as your iMac can go and the last version of Xcode supported on this version of macOS is Xcode 10.1 which comes with iOS 12.1 SDK.

You can not install any app updates from the Mac App Store that have a minimum OS version number requirement that is later that the one you have installed.


  My question is, is there any work-around that would allow me to use my current machine, but build using iOS 13 SDK?


Sadly no. It's not possible to run Xcode 11.x or later which includes the iOS 13.x SDK on macOS High Sierra 10.13.6.

Depending on your machine configuration, you may explore the possibility of running a virtual machine on your iMac that can then run macOS Mojave or macOS Catalina which in turn could run Xcode 11.x. That's the only way you can build an app with iOS 13 SDK and submit it on the App Store. But again that's neither a convenient or a future proof approach.


  I power-on this machine on three or four times per year, in order to make a tweak to the application, and then upload the new app to the App Store. I really don't need or want to buy another computer just to do this.


You can look into the possibility of renting a Mac temporarily, export your identity onto it, set it up for development, and build and submit your app update.

MacinCloud is an online cloud based Mac rental service that you can use for precisely this purpose. I have used the service in the past and found it to be satisfactory. Once you have set it up, you can start a screen sharing session to the computer and use it as you'd normally use a Mac. Check the linked website for more details.

P.S.: No affiliation with MacinCloud. Have used their services in the past and found them satisfactory.