When thunderstorms or other severe weather crop up, will the Air Traffic Control in the area proactively route traffic (horizontally or vertically) around the trouble area, or do they leave it up to the individual pilots to request an alternate route?
Yes, ATC will reroute flights that are operating on Instrument Flight Rules (IFR).
In addition, NASA is working on a product called Dynamic Weather Routes that will help controllers be smarter about amending planned or unplanned weather reroutes.
Sort of, for aircraft operating VFR.
VFR flights that are talking to ATC may get advisories about weather and suggested headings from the controller. I've received notifications about things like 'moderate to heavy precipitation 5-10 miles ahead' many times when operating on VFR flight following.
The system is a combination of both. The controller will have a view of the weather radar for the area, and can route traffic around bad areas. They will at least advise the pilots of the bad weather that they can see, and can offer the pilots a deviation or let them know what previous flights have done.
However, the pilots have a better view with the weather radar on their plane, and the pilots have the ultimate decision as to what they will do. The pilots may see a weather return they don't like and ask for deviation around it. Even if ATC denies the request, the pilots are ultimately responsible for the safety of the flight, and should do what they feel is best, even if that means deviating without permission.
There is an interesting article here that is a bit dated, but should still apply to the general idea of how ATC handles bad weather.
Pilots do not "request routes" except when entering or exiting congested air space, or Class A air space or IFR conditions. They optionally notify ATC of where they might be going by filing a flight plan. VFR pilots can also request flight following. When flight following is activated ATC will notify you of other aircraft or weather conditions that could cross your path.
In uncontrolled airspace, ATC will notify you of a dangerous condition if you are on a flight plan that would bring you into contact with dangerous weather or have flight following activated. It is up to the pilot to decide what to do with that information. In controlled commercial airspace (Class A), ATC will actively route aircraft around dangerous weather conditions.