The oscillation of species' populations due to predation.
The convergence of machine learning models using gradient descent - this is a special vector field that's tuned by uniformly multiplying the field with a scalar (bonus points for involving a segue into machine learning theory and having them tune the vector field for most rapid descent)
A couple ...
The two examples you give both have zero curl, which limits their usefulness. Examples that do have a curl would be:
an electromagnetic wave
the magnetic field of a wire, inside the wire
the magnetic field of a slab of current, inside the slab
the field of a point charge that is moving inertially.
The external magnetic field of a wire is also an ...
Air speed/direction on a weather map) is a very intuitive one. There's also other fluid velocity (and flux) vector fields in various chemE, mechE, and nukeE applications.
I personally think the air speed is most intuitive as something where you really need speed and direction (i.e. a vector, not a scalar) and it's something people encounter in daily life. ...