If Disney decides not to have them air-conditioned, no one will use them. That means they will be open to the elements and as such not run in inclement weather making them a non-practical transportation system. (Not that I would be surprised by them doing this BTW). Also super capacitors are great but tremendously expensive. Unlike London a system in Florida would need a significantly higher capacity power unit to deliver the cooling power required. Consider a 12000 BTU AC unit (and even this may not be enough cooling capacity) consumes 4-5 kWH of energy, super capacitors cost between $2400-$6000 per kWH of storage. Split the difference and say $20,000 per car in power storage alone times the number of cars they have + the actual cost of the car itself, and it adds up very very quickly.
I agree with you that A/C is a challenge, especially the weight, however it has been done before. As it has been mentioned, the Emirates Air Line in London has air conditioning. That system has the benefit of not being in the Florida sun, but other mitigation measures keep this within the realm of possibility.
kWh measures a quantity of electricity over a period of time. If an air conditioner needs so many kW per hour, the capacitor only needs to be big enough to power it for a few minutes, and wouldn't need to be as large as you propose.
Other mitigating factors include white roofs, tinted glass and well air-conditioned stations to make guests comfortable. The brochure from CWA constructions, the leading gondola cabin manufacturer, shows a wide range of vents, some specific to cabins that will be used in rainy areas.
if they can put lights and electrically actuated doors in them, they can put climate control in them.
There's a big difference between what is effectively an LED flashlight in the ceiling with a solar panel and a small battery, and a powerful A/C unit. The doors are mechanical, they are actuated by an arm when the cabin enters and leaves the station.