How do I update my Scansnap Manager so I can use it with Catalina?

As QuickTimeKirk has already mentioned, you can check with this app's developer to see if they are/will be providing an updated version that will work with macOS Catalina.

Alternatively, or if the developer is no longer providing updates, you have the option to either use another Mac with an earlier version of macOS that supports this app OR install a VM manager, like Parallels or VMware, on this Mac, create a VM (virtual machine) of an earlier macOS, and then, run this VM whenever you need to use it.

Try the vendors web page.