I'd encourage those with the skill to do so to apply for telescope time. A friend of mine, with no prior observational experience, had a good idea for a Chandra observation, applied in partnership with an astronomer, and got the time necessary to make the measurement.
It really does happen.
Things like a Sagan 'pale blue dot' image would be a longer shot, but astronomers are humans, too -- if there's a very cool and human idea out there, the committees might be receptive there. (i.e., catching the glint of light off a Mars Observer solar panel or some such thing).
My understanding is that due to Webb's location at L2, it can never point back at the Earth, because that would basically be pointing directly at the sun.
I read that as meaning "Pale blue dot" in the sense that Carl Sagan wasn't a professional astronomer or a NASA employee, he just said "Hey, wouldn't it be cool if you tried taking a photo of Earth from the Pioneer probe" and they did it.
Sagan was a professional astronomer in Cornell's astronomy department, but his rationale for making the pale-blue-dot image was less as a scientific endeavor and more as a way to tell us more about ourselves.
It also can't be pointed as the the optics need to be actively cooled, hence the giant sunshield and cooling pumps, so it'll never point towards earth/sun
Sadly seems like it's way too small.
Pioneer 10 is ~2.9m long [1] and 2.0e7 km from Earth [2]. JWST is just 1.5e6 km from Earth, so that doesn't make much of a difference.
Doing the trig, you get the arc-width of P10 is 2atan(w/2d) = 2atan(2.9m/4.0e7km) = 7.33e-14 rad.
JWST has a resolution of 0.1 arcsecond = 4.8e-7 rad [3].
So we're off by "just" a factor of 10,000,000 on resolution :-(
It really does happen.
Things like a Sagan 'pale blue dot' image would be a longer shot, but astronomers are humans, too -- if there's a very cool and human idea out there, the committees might be receptive there. (i.e., catching the glint of light off a Mars Observer solar panel or some such thing).