Why doesn't my furnace fan run in cool mode?
The blower motor works on heat, That means your problem is in the ac unit or the thermostat 

To determine if it's the thermostat you can use a couple bypass methods since the thermostat is just a bunch of hidden switches that eventually go bad, Run a piece of wire from the R terminal to the W terminal, that says you want heat on to the furnace without a switch, Then run that same wire from the RC terminal to the Y terminal, that's the method for your ac unit, if it kicked on then it's your thermostat, if it didn't then make sure you have 24 volts coming to your thermostat from the ac unit, if the ac unit don't have a transformer then you need a jumper wire from RC to R in your thermostat

Look in your ac units panel and make sure the switch is getting 24 volts on the both of the small wires and that there aren't any bugs stuck in the contacts, you should be getting 240 in and 240 out on the switch when it calls for cool air... also make sure to check the run capacitor, shut off the power, disconnecg the wires, touch the terminals with a metal screwdriver and a rubber handle to discharge it, then test it, herm to fan and herm to C for compressor, you do not want any under voltage at all, that would cause a higher amperage call when your fan is running at lower rpms 

Also if you notice any bulging or oil on the capacitor unit itself that means it's bad.