How does a layman get a 3D printed replacement part?
If you have the remaining pieces of the gear and enough remains to determine certain measurements, one can either engineer the gear using a number of gear modeling designs, or one can take measurements directly from the parts and engineer a raw design.

If the gear you have is not particularly peculiar, it is possible to use a gear generator plug-in, template, or library to make the "foundation" of your gear. The modeling software would then be used to add the appropriate bosses and key ways required to complete the design.

If you are considering to create the part yourself, you have a wide selection of programs from which to choose. I'm fond of OpenSCAD, and it does have a number of gear libraries. Simple bosses and key ways are easily accomplished using OpenSCAD.



Another package available on the internet which includes the option of using a gear generator is Tinkercad which has a reputation of being easy to use. You'll find many tutorials for this program as well.

Tinkercad requires an "outside" program to generate the gear design which is then imported to the model workspace. Even a program as simple as Inkscape can create gear profiles to be imported into many design packages.

Fusion360 is available free for hobby or non-commercial use, but may not be the easiest to learn in a short time.

So many others as well. Use your favorite search engine for "gear generator modeling software" or similar wording and be overloaded with links.