ARM M4 based Microcontroller or Particle Photon drive multiple small colour lcds?
First off it starts with pin count and the interfaces being used for these lcds. If they are spi for example, which some are that I have purchased.  Then sure, not a problem, you need between 8 and 20 pins depending on how many you can get per spi bus.  Same answer for i2c but fewer pins.  If these are lcs that take tens of pins each, then probably not you need something else or multiple microcontrollers.  which may be the solution anyway since the cost of a microcontroller relative to the lcd begins to disappear as you increase the size/complexity/performance of the lcd.

You also need some network solution as you implied you want a web server or client.  Can be solved with any of the IoT support chips these days, but not sure what you are truly after.

And you need enough storage for these photos, perhaps you can get by with only needing to store one or a fraction of one at a time as you pass it from the network packet solution to the lcd solution, or worst case you may need N lcds * 2 * biggest image if you need to have the currently displayed image plus a new image being downloaded double buffer style.