Suggestion: if you don't care about compensation, join an open source project. There are dozens of projects out there with crappy project sites. Completely revamping their public facing can draw in significantly more people (I know I've skipped past a few projects because I couldn't make immediate sense out of their site). You'll have the same effect that you're looking for here, without having to wait around for someone else to give you permission.
I agree that this is a great method for your work to get a lot of exposure, to help out a lot of people, and add something cool to both your profile and resume. It has a lot more potential than doing one free job for a company. You should really look into this.
The problem with open source projects is that it's often hard to break into their communities. Some can even be surprisingly insular, and "don't need" what you see they are obviously lacking.
Of course, if you find a good one, it can be incredibly rewarding, and you can make new friends.
That's where GitHub is really great. Find a smallish but active project that you find cool, clone it, fix some bugs or add some features, and send a pull request to the project owner.
