Issue

Several fonts that I tried with Python Pillow, don't support emoji printing.

Solution

Discord-specific standard emojis look a bit different from emojis on other platforms, I've found an API that provides those specific Standard Discord Emojis example. In lines 149-155 link to the emoji image is obtained using the hex code of the emoji.

Once you have the url to the emoji you'll need to paste it as a small image in the appropriate part of the message.
For Custom Discord emotes like :baj: I did manual in-line image insertion in lines 190-206, where I print the text leaving blank space for the emote and later insert the emote images in those spaces. You could add the standard emojis in the same way