I don’t think it would be much use for searching for it. For this you need a very sensitive camera, which the JWT is. But you also need a large field of view.
They are searching using Subaru, an 8.2 meter optical-infrared telescope in Hawaii.
JWT is 6.5 meters in diameter, so Subaru gathers 60% more light. But the main thing is the field of view. Subaru has a field of view of 1.5 degrees, so three times the diameter of the Moon. SUBARU
By comparison JWT has a field of view of 1.25' x 1.88' - that’s arc minutes. There are 60 arc minutes in a degree. That’s large compared to Hubble, which has a field of view measured in arc seconds, but small compared to Subaru.
So, it’s just not designed for the kind of large scale searches needed here. It would be sensitive enough, but you’d build it differently if you wanted a wide field instrument like Subaru. For instance, Subaru is a Ritchey–Chrétien telescope which is especially designed to have clear images off axis over a very wide field of view.
Horses for courses. You can do searches of this nature with space telescopes, and there are ideas for wide field space telescopes to search for Near Earth Asteroids, for instance. And the WISE telescope is an example of a quite small space telescope, only 0.5 meters in diameter, that was optimized for wide field imaging, and gives the tightest constraints we have on this new planet X (before the Subaru searches complete):
Artist's impression of the Wide Field Infrared Survey Explorer which has produced the tightest constraints to date on Planet X.
But James Webb is not designed for this task, as it is optimized for a tiny field of view to look very close at particular things in space, like Hubble.
It would however be useful for looking at new planets and dwarf planets in our solar system once found, like Hubble.
Also by the time it launches, we may have found this planet already (if it exists).
See also my Why This New "Planet X" Is No Threat To Earth :).
Also, just to register a token “protest” here about the name. Of course it’s natural to use it, as it has been taken up by the press and everyone calls it this. But I hope we find a better name than this at some point!
When has any planet been called by a number? Is Earth “planet 3”? And it also is a name that would change depending on changes on how you define planets, or on whether we find any other planets closer than this very distant one (well possible) and indeed, on whether it is a planet at all (it is a rather borderline “planet” by current definitions if it exists).
And I think it’s at least possible that some future meeting of the IAU could change the definition of a planet and include what we now call “dwarf planets” as a planet, or do something to change the current rather odd situation where a “dwarf” of a planet doesn’t itself count as a planet.
Especially, the current definition could get very awkward if we find objects on the borderline between a “planet” and a “dwarf planet” where some of the proposed measures of how it clears its orbit say it is a planet, some say it is not, and others just can’t decide the question without more observations. I think it’s at least possible we do find objects like that eventually. And it would be a very awkward situation to just not be able to say if some newly discovered large object in the outer reaches of our solar system is a planet or not. That could even happen with this new Planet X if it turns out to be a bit smaller than expected (as happened with Pluto). Maybe like Pluto just a coincidence that it is found as a result of a thorough search astronomers wouldn’t have done otherwise, or that it is only part of the explanation of the disturbed orbits that led to its discovery.
See Would New Planet X Clear Its Orbit? - And Any Better Name Than "Planet Nine"?
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