Yes, you can become an officer with a misdemeanor marijuana offense. While it is not an automatic bar to employment, it will be taken into consideration in any background. A LOT will depend on what you have done in the intervening years SINCE that time. A solid job history, a good track record at school, and no further criminal acts will go a long way to showing you have grown up since then. The more time that has passed, the better.

You may also look into any options that might be available in your state to seal or expunge juvenile convictions. While you would still have to admit to it in a law enforcement background, it would go a long way to clearing your history up.