This may be a blatantly obvious question but I've never heard about such a thing happening. Could this be possible in some way using a vacuum chamber to boil the water and spin a turbine?
The answer is "no". Steam turbines use hot, high pressure steam. As this steam travels at high velocity through the turbine blades, it experiences pressure drop and a drop in temperature. The resulting "loss" in steam energy is what produces the work that turns the turbine, which is connected to an electrical generator.
For water boiling under vacuum conditions, the steam is produced from the heat content of the water itself. This heat content is limited, so the amount of steam produced is limited. In addition, because the steam is produced under vacuum conditions, it has a very low density and a very low temperature, so there is not much steam energy available to do turbine work. This fact becomes even more "onerous" when one considers the amount of work that must be done by the associated vacuum pump in order to produce the vacuum conditions required to boil the water. Once the vacuum pump work is considered, it becomes obvious that the efficiency of this process is WAY too low to be useful.
Since water boils at room temperature in a near vacuum, is it possible to use this to generate electricity in some way?
Yes. If something is making water boil, then there must be a flow of heat. You can build a heat engine that extracts useful work (e.g., to generate electricity) from that heat flow.
Could this be possible in some way using a vacuum chamber to boil the water?
Vacuum doesn't make water boil. Heat makes water boil.
When water boils, the vapor carries heat away. If you suddenly expose an open container of water to vacuum, two things will happen: (1) the water will start to boil, and (2) the water will get colder as it boils. When the water gets cold enough---maybe cold enough to freeze solid---the boiling will stop.
In order to continuously boil water, at any pressure, you must have a continuous supply of heat.
So, what you didn't ask was, would pulling vacuum on the outlet of a steam turbine improve its efficiency?
Well, here's the kicker: In large thermal power stations, that is what they actually do.
The working fluid in a large thermal power stations is de-mineralized water that flows in a closed loop. There is nothing but water in the loop (i.e., no air), so at the cold end of the turbines, where the temperature may be as low as the temperature of the cooling water that they draw from a nearby river, the pressure can be much less than atmospheric---almost vacuum.
In the function of a machine, no.
Some biophysics people propose that varying degrees of vacuum pressure may occur in certain confines molecular spaces in tissues. E.G. this is a proposed mechanism for how tall trees, e.g. redwoods, are able to draw water up so high. (because water evaporates from the leaves, and that leverages a vacuum force between the chain of water molecules that goes all the way to the roots.)
So in the above case, a biological vaccuum may be at work, using energy that would otherwise be lost to thermodynamic noise. This is not "generating electricity" per your question, but it is 'generating' energy.
The second law of thermodynamics requires that for a heat engine to generate electricity, it must have a heat difference to exploit.
You could probably use the fact that water boils at low pressures to make a steam engine that ran on the difference between room temperature and 0 degrees C. But the pressure in the condensation chamber would have to be even lower than the pressure in the boiler, so the engine would probably have a very low power output.
If you were actually designing an engine to run on that kind of temperature difference, it would be better to use some working fluid other than water that could boil at room temperature, even under a high pressure.
Note: There is a reason why the water-based engine could not be run with a cold temperature that is much cooler than 0 degrees C. The reason is that the triple point of water is located at 0 degrees. So if you tried to keep the condenser much cooler than that, you'd have solid ice forming in your condenser: not a good thing.