Where are emojis stored on Macs?
Emojis are handled like any other fonts, the corresponding font file is stored in /System/Library/Fonts/Apple Color Emoji.ttc. So there are no images per se which you could reuse (even ignoring any potential copyright problems).
Three options on how to get emoji images (without programming a little tool to parse the font file directly):

use the Character Viewer as described in this answer and a screen grabber to get an image from the emoji you are interested in
use Font Book to look at the characters/glyphs available in &quot;Apple Color Emoji&quot;
use Google Image Search to find bigger image file representing the emoji you need