I'm thinking of something like ZenDesk and other offer with a very simple script that gets embedded and creates a floating 'help' button on every page of the website without having to code any site specific forms.
There is no widget like this in osTicket, but its pretty easy to make your own.
If it is pretty easy, it would be fantastic if there was an out of the box solution as part of the distribution that could be customized!
So then make it up, and post it for the community to use.