How can I have an exhaust fan run if either of two switches is on?
This is the fundamental circuit you need. Use two double-pole switches. One pole of each switch controls one damper, and the other poles are connected to the fan in parallel.

If you need to control the fan and dampers with timers, humidity sensors, occupancy detectors, etc., which rarely come in double-pole versions, you will probably have to use relays in addition to the switches. The relays will, in turn, be activated by your single-pole detectors.

The physical wiring will depend upon the locations of your switches and loads, how you choose to run your cables, and where power enters the circuit.
Here is an example with power entering at one switch, the fan cable branching from the other switch, and no cables between the loads.

I have omitted the fault ground wires from both diagrams. They clutter the picture and make it harder to understand. Just connect all fault grounds together wherever you see them.
Many many other physical layouts are possible. I think any layout that implements the fundamental circuit and does not have any cable loops will be safe. Once you decide where your cable runs must be then you can ask for a specific diagram.
One other possibility is to use the dampers as relays. You will need a style of damper that incorporates a switch to indicate when it is open. If you can find (or safely make) such a thing, you can wire those switches in parallel to control the fan.