Do any programs need to be installed that aren’t included in the default install of Linux Mint in order for it to allow VNC connections?
Yes, you’ll need a VNC server. I don’t know if Mint includes one by default, but popular choices are Vino and RealVNC (Xvnc).
The way to do it is go to menu->control center->remote desktop. Then click on allow others to view your desktop. It will also show the ip that people need to enter to connect.
I have a headless Linux Mint server that uses vino for VNC access. I configured it to autologin, and then I access the server with VNC whenever I need it. It works great.