If you're using a keyboard and accessing Mastodon through your server's website, you can type emoji without having to use the picker.
Just type a colon : and without any spaces start typing a keyword. It will then present you with five emoji that match the keyword, and you can choose the one you want with the arrow keys.
For example if you type :sun you will see five different emoji for "sun".
If the one you want isn't suggested, click on the 🙂 picker icon instead.