If a camera truly has no focusing mechanism, it doesn’t refocus at all: it uses a fixed-focus lens. These are usually designed with a small aperture and a very small sensor, which creates very large depth of field, so a wide range of distances looks acceptably sharp.

They are often set near the hyperfocal distance rather than simply at infinity. That way, everything from some nearer distance out to infinity is reasonably sharp. The tradeoff is that very close subjects may fall outside that acceptable range, so yes, fixed-focus designs usually have a minimum distance where sharpness starts to look acceptable.

Also, not every small camera is fixed-focus. Some webcams and compact cameras do have autofocus and physically move a lens element using a tiny motor or similar actuator, with focus determined electronically or by software.