Yes, with interferometry based synthetic aperture imagery, but...
It would take several orders of magnitude improvement in digital signal processing technology for it to work at the scale of a video screen. The sensors will need to record at a very high data rate, on the order of petahertz.
Every single pixel will be receiving all of the light that is in its field of view, and none of the light will be focused. Fortunately, light isn't just photons, it's also electromagnetic waves. The sensors will behave like antennas, rather than eyes, passively collecting the radio waves at the wavelength of visible light.
Radio signal sources can be tracked quite easily, by measuring the timing of a wave reaching multiple antennas. The same applies to all sorts of waves; an earthquake's epicenter is calculated by measuring the timing when the waves arrive at seismometers.
We even have the same types of sensors built into our own human bodies. : Our ears can taking a wave and determine the direction that it came from, based on changes of intensity and timing.
We have already proven the concept using vast antenna arrays to collect unfocused photons, to create a focused image. We imaged the supermassive black hole M87* using the Event Horizon Telescope.
Of course, when astronomers use the term "photon" they don't just mean visible light; they mean any coherent electromagnetic wave. This image represents the peak of current engineering feasibility for synthetic aperture imagery. The EHT uses a Very Long Baseline Interferometry, which works in 450 GHz, using very narrowly calibrated equipment designed specifically to tease out the glow of the accretion disk at the wavelengths to detect event horizons around black holes.
In order to get meaningful data, though, your sample rate needs to be at least twice the frequency of the signal rate, preferably more than 4 times the signal rate, or you start getting downsampling errors called aliasing.
In order to record visible light, which has frequencies between 405 THz to 790 THz, you will need a sample rate that is at least 1.58 PHz.
Due to limitations from the speed of light, and the time that it takes electrons to pass through silicon and copper in computers, this is just past the fastest speed that we can record data meaningfully. We would have to pass the data from several sensors in order to build up a meaningful synthetic aperture image from interferometry. We would need specialized recording technology that we just don't have yet.
And, there's also the problem that LEDs aren't designed to collect light, even though they're capable of doing it... just as sound coming out of a microphone would sound terrible, and sound recorded from speakers is also low quality.
It would take several generations of iterating on the current science in order to use an LED-based computer monitor to record what's happening it a room, and it will always require specialized systems to just record the data in a meaningful way, much less process it into an image. It took several petabytes of data and 3 years of processing in order to build the EHT's image of M87*. It was faster to hand-carry the hard drives from the telescopes around the world to the datacenter, than it would have been to send the data over high speed internet links.
It would take a lot of iterative work to miniaturize the chips necessary to do the calculations, but the technology just barely exists. Such a screen would be prohibitively expensive, as it's much easier to just put a lense in front of a cluster of photodiodes (i.e., a webcam) and hide that in the corner of the screen, but those leave physical evidence... you can see the lense if you take the screen apart.
It could be possible, with non-digital interferometry, to construct such an image of the room in real time... but that equipment barely fits in the basements of large telescope observatories, and requires cryogenic cooling. You wouldn't be able to collect it clandestinely, and would be much cheaper (and higher quality) to use a webcam.