Where can I find a version of iWork that runs on Catalina?

You do not search for Keynote in the App Store, but in your Account page.

In Catalina, you select your account in the bottom left of the window.

Then - assuming you have downloaded it before - it will show in your purchases:

The current versions (11.2) of Pages, Numbers and Keynote in the Mac App Store now require macOS Big Sur or Monterey, and no longer can be installed on Catalina as could versions 11.1. Apple does not provide downloadable, full installers for the previous versions of these applications.

Now, you will need to use the iCloud versions of these documents, or if you have iOS/iPadOS 14.8 - 15.*, you can use Pages, Numbers, and Keynote on that platform.

By the way, Apple last used the iWork brand name for the iWork '09 suite of applications, and since Oct 2013, has used the application name with its current version as a brand reference.

If the three application were Previously Owned and associated with your Apple ID - check in Previous Purchases in the Apple Apps Store for them. The current versions are version 11.1

Worst case - head over the www.icloud.com and log into iCloud with Apple ID and Password. The three applications can be used from there.

Are you sure you were trying to download from your purchases and not the general page in the App Store?

Did you click your account at the bottom left and see all the applications that you have purchased over time, and then click on the download button next to, say, Keynote?

By doing that, it has been the general experience to get a dialog to the effect that "the current version is not compatible, do you want to download the most recent version compatible with your OS" (not the exact wording but that's the gist of it).

Note: I don't have access to a mac running Catalina at the moment, so I cannot test, but I have been through the above experience several times in the past.

Sometimes when I had ( past tence ) Catalina on Antiques iMac 2012 - the previous purchased may or may not work for the former iWorks Applications. Control Click the Pages Application in previous purchased - if lucky, a pop up appears and states something like not compatible but will allow the download.

When the Pages, Numbers, and Keynote product teams add features uniquely supported in Big Sur and the forthcoming Monterey, then they up the minimum operating system support version. Has nothing to do with the hardware, unless that hardware cannot upgrade to that minimum operating system requirement. Nor is this a first time that Apple has upped the version and minimum operating system requirements for Pages, Numbers, or Keynote.

This is the downside of using these applications, instead of third-party solutions that are not as sensitive to Apple's own application requirements.

Sure, they do work and are a valid alternative - though "easier" is not a word I'd use.

Anyway, I am glad you found a solution, but still I'm puzzled, as I can download old Pages, Keynote and Numbers on a mac running High Sierra. I just confirmed a minute ago. I clicked install and got "The current version requires 11.0 or later, but you can download...". It worked just fine.

Slowly work on the level thing. :-))

Welcome and very best