Is it realistic to build a telescope powerful enough to take a photo of TRAPPIST-1's planets? Assuming modern and near future technology, could we build a telescope powerful enough to actually see TRAPPIST-1's planets?
It's not out of the question, but it would be very difficult to use direct imaging for any of the exoplanets around TRAPPIST-1:


The exoplanets are small, all not much bigger than Earth. Ideally, you want a target the size of Jupiter or bigger. Likewise, they are very low-mass, and are thus small. Once more, this is a problem. Direct imaging has only had much success with large planets, of many Jupiter masses.
The exoplanets are close to the star, with the furthest one out - admittedly with a poorly-constrained orbit - having a semi-major axis of 0.063 AU (and a year of 20 days), give or take. That is tiny. Mercury's semi-major axis is more than six times that. If the exoplanets are close to the star, it makes it much harder to distinguish them from the star's glare. Granted, there are methods that do this very well, but they're not meant for planets this close.


To see what I'm talking about, look at some examples of directly-imaged exoplanets. Almost all in that list have many times Jupiter's mass and at least its radius. Their semi-major axes are many, many - sometimes thousands - times that of Earth.

I can't speculate as to what technology we'll have decades from now, but if you want to find out more about the exoplanets around TRAPPIST-1, direct imaging is not the way to go, and I suspect it will not be a very popular choice among astronomers.