Would it be possible to acquire the energy to move the heat from the
room-to-cool itself?
If I understand you correctly, your proposal would look something like FIG 1 below. It consists of a heat engine that takes in heat from a singe thermal reservoir (the environment being cooled by the refrigerator) and produces work to drive the refrigerator that cools that environment. There is no violation of the first law (energy is conserved), but producing net work in a cycle while transferring heat with a single heat reservoir violates the Kelvin-Planck statement of the second law which is:
No heat engine can operate in a cycle while transferring heat with a single heat reservoir.
Some heat must be rejected to an even lower temperature environment. In addition, since the net work done in FIG 1 is zero (output of heat engine equals input of refrigerator) FIG 1 is then the equivalent of FIG 2 which is a violation of Clausius' statement of the second law which is
No refrigerator or heat pump cycle can operate without a net work input
Meaning, you can't transfer heat from a lower temperature environment to a higher temperature environment without doing net work on the system.
Or would it even be theoretically possible to not move the heat
energy, but instead convert it into e.g. electricity?
Again, you can't take heat from a single thermal reservoir and convert it completely into work in a cycle as it is a Kelvin-Planck violation. Some of that heat must be rejected ("moved") to a lower temperature reservoir.
I can imagine a ventilation system that takes in warm air, outputs
cool air, and converts the energy delta into electricity to insert
into the national grid.
Ventilation systems (intentional types) do work to move air, they don't convert heat to work. What you are describing is a heat engine which takes in heat from a high temperature environment, produces work, and reject heat to a low temperature environment. The work produced can, of course, be used to generate electricity.
Hope this helps.
