Instagram uses the standard emoji keyboard built into iOS or Android. When in the Stories interface, swiping up from the middle of the screen reveals a set of stickers, and under that, recent emojis.
To pin an emoji to a story — where it will stay on the object you choose, even when it moves - first insert the emoji or sticker, then press and hold to bring up the pin interface.
Instagram permits browsing photos via emoji hashtags. Examples: