Simply speaking: EmojiOne is dead. They removed the CC-licensing option from SVGs with version 3 and later did the same even for lower-res bitmaps.
The current release (renamed JoyPixels) has a strict non-commercial rule even for lower-res bitmaps, rendering the entire project incompatible with free software: https://d1j8pt39hxlh3d.cloudfront.net/contracts/finalized-pdfs/free-5.0.pdf
Twitter is maintaining a fully free emoji set that has a similar design as EmojiOne: https://twemoji.twitter.com/
The designs are under CC-BY 4.0. They are quick to release new versions with each Unicode Emoji version bump every summer.