My house uses two Maple Chase Thermostats to separately cool the upstairs and downstairs from one HVAC unit. Can anyone tell me how to run the fan only with this arrangement? It doesn't come on even when both switches are set to fan.

You have a zone system. The fan is not designed to run unless one of the zones is calling for heat or cooling because you have dampers that control the air flow. Usually these systems are chock full of low voltage wiring and it will be difficult to run the blower for both systems unless a electrical by pass switch is installed which will activate bothe zone dampers. Best left to a professional on site so they can see your wiring and which zone brand you have.