Most problems with Local Storage happen when people are trying to access data before it's loaded. Or writing and reading data simultaneously in one tick. If you avoid making these mistake, then LS should work just fine.
Unless your JSON data is very large, saving/loading should take a fraction of a second. You can add a wait of 0.5-1 seconds, but to be safe use "LocalStorage On exist" or "LocalStorage On set" events. For example, if you need to save to LS and switch to another layout, move "Go to layout" action into the "LocalStorage On item set" event.