Is there a way to update the emoji font?

The font file you need to use a more recent version of is /System/Library/Fonts/Apple Color Emoji.ttc and is about 135MB.

The High Sierra version of the file and the process to install it to /Library/Fonts can be found here: http://stephenradford.me/install-high-sierra-emoji-on-older-versions/

Note that the Emoji & Symbol picker will not list the newer emoji added since your version of macOS. For that to work you would also need to update at least /System/Library/PrivateFrameworks/CoreEmoji.framework and maybe some other files.

You have to either obtain a copy of the newer Apple Color Emoji font from a newer OS version and install it in your machine, or manually edit the one you have using a font editor app which can handle this kind of font. I think the second option is not very practical for most users in terms of the work and skills required.

If you're on El Capitan, it's quite easy. You need to temporarily disable SIP and copy the TTC file over to /System/Library/Fonts. You should turn SIP back on after you copy the font, since that imposes a security risk, since any app can have unlimited access to the /System folder once you grant it access to file system.

Yosemite and earlier are the easiest. All you need to do is copy the file to /Library/Fonts, and maybe reboot for the changes to fully take effect.

Catalina is slightly different. Once you disable SIP, you need to remount the /System partition as Read/Write (again, another security risk) by typing sudo mount -uw / into the Terminal. Copy it to /System/Library/Fonts

With Big Sur, you can't replace the font. However, it's still possible to update the font. Simply install the font, and disable the preinstalled font through Font Book.

Make a back up of the original font!