Say there's an exoplanet identical to Earth 20 light years away, and also assume it orbits at an angle so that it transits its star when seen from our solar system. Would the JWST be able to detect chlorophyll on such a planet?
Detecting chlorophyl on exoplanets is definitely something people are interested in: see this paper (published as "Detecting biomarkers on an exo-Earth", Timothy D. Brandt, David S. Spiegel, Proceedings of the National Academy of Sciences Sep 2014, 111 (37) 13278-13283; DOI: 10.1073/pnas.1407296111), for instance. On the assumption that people are writing papers in refereed journals discussing the design of telescopes to do this I assume it is something that a plausible telescope could do. I am not competent to technically judge this paper however.
Here is an extract from their conclusions (omissions indicated by [...]):
We find that a future space mission will be likely to detect water on an Earth twin with a spectral resolution of $R \gtrsim 40$ and a SNR per bin of $\gtrsim$7. Such a mission will have a much more difficult time detecting atmospheric oxygen [...] For a mission targeting only O$_2$, we find an optimal resolution of $R \sim 150$ for our intermediate noise scaling case, and a minimum SNR of $\sim$6 at $R = 150$. This is $\sim$3 times the resolution of an instrument optimized to see water, and a factor of $\sim$2 more challenging than water as measured by the scaled SNR.
Finally, we show that the "red edge" of chlorophyll absorption at $\lambda \sim 0.7$ $\mu$m will be extremely difficult to detect, unless the cloud cover is much lower and/or the vegetation fraction is much higher than on Earth. [...]
Based on our findings, we argue that a future mission should be designed towards the well-defined goal of sensitivity to O$_2$ and H$_2$O around the best candidate terrestrial exoplanets, perhaps even with two dispersing elements to achieve both $R \sim 40$ and $R \sim 150$. Extensive (and expensive) follow-up of the very best targets, preferably with O$_2$ and H$_2$O detections, might then be used to search for the red edge of chlorophyll.
Similarly I assume that, since the above paper doesn't describe doing this with the JWST, this is beyond its capabilities.
This is still a bit vague - it's tough for a satellite to detect plants (with or without specifically looking for a chlorophyll signature) at low spatial density in a desert. If your hypothetical exoplanet were covered in a thick layer of cyanobacteria the signal would be much greater than the signal from a copy of Earth.
However, AFAIK the best we've done so far is to make some guesses about the atmospheric content of some exoplanets, based on absorption spectra. I don't think we're anywhere near the resolving power required to observe a planet in "first/last - quarter" phase as would be necessary to see an illuminated portion of the planet's surface.