I think the JWST is to far-sighted to be very useful for spotting something in an Earth intercept orbit. For one, it is to the outside of Earth's orbit and its instruments work in infrared bands requiring extreme cooling--it can't be aimed at angles towards Earth's orbit because of solar radiation. The shield at the "bottom" of the telescope allows for some inclination but not a tremendous amount.
No, it can watch asteroids, kupier belt objects, etc. Sure, not ones already near earth orbit, but most objects are further out at least at some point.
However as far as I know it's unlikely to actively survey them. It's expensive and limited time, so it'd focus will be on what can only be seen in infrared, like the early universe and exoplanet spectra. But who knows what the scientists will decide look at.
Hum... I don't think anybody expects objects on an orbit near or more internal than the Earth's one to intercept us. The Solar System is quite old, and any nearby object would have already hit us.
All the remaining objects that can intercept us stay far away most of the time.
https://webb.nasa.gov/content/about/orbit.html