Start Small and Work Up
This is a great idea both for your company for the reasons stated and for yourself as a learning/growing experience. When you can pull off a project like this well, you'll be ready to move up to a job with more responsibility and you'll have successes on your resume to justify the move.
Two caveats to watch out for:
- It's probably going to be harder than you think. Coordinating software development projects well has surprisingly little to do with writing good code. It's a new skillset, and not a trivial one to learn. Don't take on a hard project as your first 'team lead' experience, much in the same way that you wouldn't write a really complex application for your first program.
- Watch out for management to be concerned that you are overtaxing company resources or distracting employees from their day job. It's pretty easy to have something like this boomerang on you to be perceived as a negative. A great way to deal with this is to 'time box' the work so that it's guaranteed to not exceed what management is comfortable with giving. At my workplace, we have a 'Day Of Caring' where every employee is encouraged to take a work day off to do whatever they like at a local charity. It's a great program -- and it's very explicitly limited to one day only.
With this advice, you may want to find a very small project that can be accomplished by 2-3 people in 2-3 days for your first go. If I were you, I would consider doing it informally as a weekend activity with co-workers you have a strong relationship without even discussing it with company management. After you have 1-2 of these very small projects under your belt, you can consider moving up to more formal and larger projects.
One final piece of advice from someone who does a lot of on the side little projects for charities -- consider the 'maintenance' part of your work. Especially if you're going into an organization that has no 'it guy', you're going to find that whatever you do needs some level of maintenance over time. Even something as simple as setting up a free hosted website takes a couple hours a year to update, etc. Once you donate your time to create a project, most charities will assume you've also signed up to maintain it. Think this through before getting yourself over-committed to something you can't maintain.