There is no built-in support for any cloud system.

In help of UltraEdit there is the page INI File Selection and Advanced Settings. It explains in which order UltraEdit searches for the user configuration files. If all computers are in same network, the best solution for you might be using UEINIDIR environment variable and have all the configuration files stored on an always running computer or NAS in network. The default storage location is %APPDATA%\IDMComp\UltraEdit.

When not using Windows registry for settings, it is possible for every UltraEdit user to upload this directory into a cloud and download it from cloud on another machine. If %APPDATA% expands to different directories on different machines, it would be good to update some paths in uedit*.ini before starting UltraEdit with configuration files updated from cloud with the script which loads the configurations files from cloud.

Hint: I would strongly recommend on uploading the configuration files to a cloud to pack the files before upload for example with WinRAR using full encryption and a long passphrase as some files like uedit*.ini could contain confidential data (file names, search and replace strings, ...).