I have a website that works perfectly in any browser.
Now I want to use the Web@Work browser from MobileIron to use this website as a boorkmark to create a basic iOS application packaged thanks to MobileIron.
The basic presentation of Web@Work is here: https://www.mobileiron.com/en/products/webwork
I would like to find a way to develop with Web@Work locally, like that I would be able to track bugs easily.
Web@Work is a dual-mode app. This means that if it is installed on a device with MobileIron infrastructure, it uses MobileIron for security and management (AppConnect mode). If that infrastructure is absent, it works in standalone mode (non-AppConnect mode) and it is not secured or managed by MobileIron.
If I understand your question correctly, you're making a site and want to verify that it works properly with Web@Work. You should be able to do that by just installing the browser app from iTunes and browsing like you would with any other browser.
I think that it is not possible this way. If you would like to access an intranet web site behind a MobileIron Sentry and want to debug the traffic, you have to build your own custom browser with MobileIron SDK included. I assume there is nothing really special in the Web@Work Browser, because it is distributed via Apple AppStore, therefore they can't do really magic things inside and also have to use UIWebView for rendering web pages.
Apple does not allow to attach and debug(with a WebInspector) a UIWebView present in AppStore App.This is a apple limitation and the same applies for any other browser dowloaded from AppStore not just Web@Work.