Most of the work for this is likely to be following the "new application feature tutorial" (http://zulip.readthedocs.io/en/latest/new-feature-tutorial.html) to add a new administrator setting to control this feature (since some realms will want it to be admin-only) with a unique name like emoji_add_requires_admin (probably can come up with something clearer).
To implement the feature itself, I think it makes sense to factor the "Realm emoji" configuration options to be their own Handlebars template rather than part of the admin_tab.handlebars file, and include that template in either the #settings or #administration page, depending on the value of the user-facing setting. (Actually, I guess we probably want a read-only view under #settings even if emoji_add_requires_admin=false, so maybe we'll include it in both with the option controlling whether it shows the "add" UI or just a notice that users should add their own emoji)
@porterjamesj as FYI in case you're interested in working on this :)
emoji: Add add_emoji_by_admins_only realm setting.
This setting controls whether normal users can add realm emoji.