With the emote as you define it in your question (as opposed to an emoji or smiley, which is a different form of emoting), and in a gaming context you will encounter that word directly borrowed from English, either as emote or sometimes émote. Because it's a non-standard word, the gender of emote will be up to the speaker. I would lean towards une emote (because une émotion, une animation), but I've seen and heard un emote used before.
If you're looking for an established French word in a more general context, animation will probably be your best bet, an emote here being a character animation.
Source: personal experience. In general, with multiplayer games pitting you with international players, the English word will usually impose itself as a standard across languages.