Become an Umpire
Talk to umpires after Little League games. Tell them you are interested in becoming an umpire. Most umpires will respond by either giving you the right contact information or passing on your information to the appropriate contact.
Talk directly to the Little League. Tell them you are interested in volunteering as an umpire for the league. Little Leagues are always looking for new staff members and will probably not turn a volunteer away at the door.
Pick up a volunteer form at your local Little League or visit the Little League website for a printable version. See the Resources section below for a link. Consider this form an application. You will need to approved before you get started in the field.
Umpire for Local Little Leagues
Buy a couple books on the subject. Baseball is a more complicated game than we sometimes remember. There are rules in the rule books that are rarely ever needed, but can still come up in a game when you are on the spot.
Prepare for your first game. A lot of new umpires underestimate the job. Umpiring is a lot of work and includes a lot of responsibility. Don't take it lightly.
Find a more experienced umpire to show you the ropes. Every Little League will have veteran umpires around that will probably be happy to teach a rookie. A lot of professional umpires have had mentors in their time.