You do not have to logout for changes to hotbars to be saved, they will be saved to disk as soon as you change them
Click the browse button below and locate your appropriate HOTBAR.DAT
On windows these are in
%USERPROFILE%\Documents\My Games\FINAL FANTASY XIV - A Realm Reborn\FFXIV_CHR...
If you have multiple characters, you will need to find the write character data that you want to configure One helpful method of doing this would be to look at the Time Modified of its appropriate files, if you're working on it now, it should have the current time on it
Files are not uploaded anywhere, modifications are made with the security of your browser
Click yes (or no) to make apporpriate configurations to your hotbars
You will be finally asked to save your new hotbar configuration, replace your existing HOTBAR.DAT with the new file
Login to your character again and see changes made