It's a little request, I use Firefox dev tools for debugging my web projects and there is no scrollbar in 'Parsed Value' the viewer of local storage. So where there is large JSON to view i have to decrease the zoom level of the browser to completely view it.
I have exactly the same problem. This makes it very difficult for me to use this tool in the way I imagine it was intended. Instead, I'm generally grabbing what's in the "value" field and assigning it to a variable in the console to look more closely. Or perhaps there's a setting to add a scrollbar which I've overlooked?