To change your Moji on the website, Tap “Profile” in the upper-right menu, then “Edit Profile.” You’ll see your Moji in the yellow circle next to your profile image. Click the yellow circle to open the Moji selector. Complete Crew Missions to unlock other Moji's. 
On the app, tap Messages in the footer, then tap the pencil at the upper-right corner of the screen. From there tap on your Moji and it will present a screen showing your currently available Mojis. Just tap on your desired Moji and you’re done!