DISCLAIMER: I am not one of the developers.
As far as I know, no special permission or approval is required to code.
Your best bet is to pull down a copy of the code from the GIT repository and get it to compile and run locally.
FIX BUGS
Then you need to look at the Mantis Bug Tracker issues and see what bugs need to be fixed.
I would suggest working with older bugs or feature requests.
Try to replicate the bug. If it still exists, make a note on the bug that you are looking into it, then check the code for problems and try to fix it.
CREATE FEATURES
Alternately, You can also check out the dev wiki and see what people want coded for the game and try that.
SUBMIT BUG FIX OR FEATURE PATCH TO MANTIS
Once your code is ready, there are instructions somewhere on how to create a patch that you can upload to the specific mantis bug.
A programmer who has access will then take your patch from Mantis and try to apply it.
Repeat.
Start
here.
Once you earn your wings by coding acceptable stuff that is used for the current (Trunk) development build, I'm sure the experienced developers in the project will add your name, etc.