With over 200 mounts to collect in Final Fantasy XIV, it makes sense that some of them might fly under your radar. The Antelope Stag is one that is easy to forget about, thanks to its simplicity. Luckily, it's an easy one to add to your collection.
While it's not cheating per se, there is a way to get the Antelope Stag with little to no effort at all. The Antelope Stag Horn isn't market prohibited, meaning you can simply purchase it on the market board from anyone that happens to have it listed. Compared to other mounts available for purchase, this one tends to go on sale for pocket change. At the time of writing, the prices range from a thousand to a few thousand gil on the North American Data Centers. Even new players shouldn't have an issue collecting that much gil.

If the prices aren't that low for you, either because you're on another Data Center or because the mount has gone up in value, the proper way of obtaining it isn't too much harder. The Antelope Stag was added as part of Patch 5.5, which completed the Ishgardian Restoration and introduced Fêtes in celebration. The mount can be purchased with Fête Tokens or randomly given by Fête Presents, which are obtained by participating in Fêtes.
After unlocking the Firmament, Fêtes can be joined in much the same way as Fates. Once you've entered the Firmament, you should be able to see a timer above your quest list that shows how much time is left until the next Fête commences. When the time comes, you'll see a dialogue bubble on your screen instructing you on where to go and how to participate in each minigame. So long as your current class is a Disciple of the Land or Hand, you'll be able to hit the "Join Fate" button and start playing. Fête Presents are rewarded at the end of each minigame, and the number earned will depend on your performance.

You might get the Antelope Stag Horn from one of your Fête Presents, but it's more likely that you'll have to gather up 30 Fête Tokens from your presents in order to buy the mount. It can be purchased from Enie, one of the NPC merchants in the Firmament (X: 12, Y: 14). Once you've put in the work (or snagged it from the market board), you'll be one mount closer to completing your collection.