I'm scott_ev and I've been a member of the Arizona Local Community Team since Feb. 2009 and an Ubuntu Member since July 2010. What I'm going attempt here is to walk you through the process of getting ready to apply for membership, and if I don't leave anything out, you should be successful on your first attempt.
The first thing to do is to setup an Ubuntu Wiki so you can document everything you do for the LoCo and for Ubuntu in general. Here's a link to mine - https://wiki.ubuntu.com/j.scott.gwin . I recommend looking at several wikis and find a format you like. Once you have a basic wiki up you'll need to generate OpenPGP keys and add the public key to your wiki and sign the Ubuntu Code of Conduct.
Keep in mind that it's vital that you find something to contribute to Ubuntu in general aside from your LoCo involvement. For a while I chased bugs as a part of the Bug Squad and I helped with the Ubuntu Weekly News. Helping with the UWN was a great way to volunteer as it didn't take up a lot of time and helped me build my wiki. There are many things you can do – you'll just need to think of something. If you are a developer there are even more options available.
That's the bare bones basics. I'll try to follow this up with more detailed instructions on OpenPGP, etc. If I can help find me in #Ubuntu-US-AZ and/or email me at j dot scott dot gwin at ubuntu dot com and I'll find someone that can actually help you.
Here are a few links that might help: