I'm not aware of any single furry emoji, but I've noticed a lot of furry communities on the Fediverse using custom (non-Unicode) emoji packs; see [1] for how that might work. Also an example of the feature in use[2].

[1]: https://misskey-hub.net/en/docs/for-users/features/custom-em...

[2]: https://thegayagenda.fans/notes/ah5imm41r1ru05a7