Photography tricks for widening objects or reducing narrowness?

What you are describing is the effect of perspective, which ultimately comes down to distance.

Regardless of sensor size or lens focal length, perspective is determined by the distance between the camera and the subject. If the subject is near to the camera depth is exaggerated, and faces will appear narrow and angular. Faces further from the camera will appear flattened and more broad, and in general people find this presentation more flattering.

This is entirely a function of distance, not focal length ("zoom"). A subject which is distant but small in the frame (for example, a single face out of a group photograph) has the same perspective, and thus the same proportions, as a subject at that same distance photographed using a telephoto lens to fill the frame.

So what you need is not a specific camera, or a specific lens. Rather, you will want to put as much distance between yourself and the camera as is practical, to achieve the desired perspective, and then use the lens that allows you to achieve the desired framing at that distance.

Your face is your face. It's not to narrow or to wide. It just is.

But, thoughtful lighting and wardrobe and framing will go much further toward allowing you to convey what you want to convey than a particular lens.

A lot of people are self conscious in front of the camera. Practice is pretty much the only way to get over it because practice is the only way to learn what looks better and worse on camera.

We all have a better and worse side of our faces.

A place to start:

1. Find your good side.
2. Key light on your worse side.
3. Fill light on your better.
4. Shoot from the fill side.
5. Experiment with the camera above or below your eye line.
6. Crappy cameras like a laptop webcam benefit more from good lighting.

It's not magic in a box. It's work. Some physical. Mostly psychological.

Yes, there is a very simple way to do this: stretch the image along the horizontal axis while not stretching the image along the vertical axis. You can try this on a photo in any photo editing software. You'll discover that you can choose how 'wide' you want to look to your taste. In order to implement this, you'll need a way to do it in the context you're working in, likely using some kind of digital manipulation or filter. Can also be done using optics, but considerably less straightforward to find the equipment than to find processing software.