By default for v10.10 the toolbar configuration is stored in the registry as written above and you should have write access to it (current user key).
But if on the General configuration tab near the bottom the option Enable old style (pre-v9) toolbars and menus (quicker load times and more compatible with some screen readers is checked, than the toolbar configuration is stored in uedit32.ini and you have to use the Toolbar tab in the configuration dialog to modify it.
To use a local uedit32.ini (and other configuration files like uedit32.kbd) create in the user environment the environment variable UEINIDIR with the full directory path where you want the UltraEdit configuration files or use the command line option /i="path to ini\uedi32.ini" at the shortcut. The command line option will not work when you open a file with UltraEdit by double clicking on the file, if you not modify the command line for this file association too. So the environment variable is the better solution for you.
For more details read in help of UltraEdit the page INI File Selection and Advanced Settings.
Best regards from Austria