Internet Explorer 6 has been around for ages, and the vast majority of websites and web applications work correctly with it.
In order to change to a different browser (which, BTW, will probably be Internet Explorer 7), you need to, first of all, do a cost benefit analysis to justify this decision to everyone's bosses.
A really high level summary:
Cost:
- Test all web applications.
- Check all third-party websites actually work in the new browser.
- Check all other software, to see if there are any underlying issues with moving to a new browser.
Benefits:
Honestly, I can't see this happening, unless a corporation is doing a desktop refresh which happens periodically.