Great news! The EndNote X9.3.1 update for macOS Catalina compatibility is here.

Via EndNote email Sept 30, 2019

For those who already have EndNote X9:
If you are planning on installing macOS Catalina when it is released in October, please install the X9.3.1 update first, to avoid interruption in your workflow.
To learn more about the updates to EndNote X9, please visit https://support.clarivate.com/Endnote/s/article/EndNote-for-Mac-macOS-Catalina-Compatibility?language=en_US.

For those running EndNote X8 or earlier:
We recommend upgrading to EndNote X9 as your current version will not work with macOS Catalina.
Alternately, you may keep working on your current macOS version.
For more details, please visit https://support.clarivate.com/Endnote/s/article/EndNote-for-Mac-macOS-Catalina-Compatibility?language=en_US.