emojidex provides a full set of emoji images which are free to use, and come with an API for developers looking to integrate in their projects in what the developers call their emoji as a service platform.
Starting today, Emojipedia lists emojidex alongside other emoji vendors.
Character support includes emojis from all versions of Unicode, including the forthcoming Unicode 9.0:
Above: Rolling on the Floor Laughing (ROFL) is new in Unicode 9.0.
Additional emojis additional to those available as part of the Unicode Standard are available as user-contributed images on the emojidex website.
Capitalized as "emojidex" with a lowercase 'e' out the front, I am told 😬 ↩