How might wind turbines affect the climate?

"It occurred to me that there is only so much energy in the atmosphere and if we capture a percentage of it with wind turbines, we are lowering the total energy of the weather system." -- The energy extracted by wind turbines is an extremely small fraction of the total wind energy on the planet. Remember that the main reason for wind is because Earth is spinning. A full revolution of a planet in a day is a lot of inertia.

This doesn't mean wind turbines don't have any disadvantages. One problem with them is that they disrupt the local airflow, which hurts the local ecosystem and kills birds.

"But I keep thinking less cooling breezes moving horizontally will create more vertical air movement rising up to form thunderstorms." -- The horizontal movement of breezes actually helps induce vertical motion to form convection.

The amount of energy taken from the atmosphere to turn turbines is extremely tiny compared to the actual energy contained in the overall atmospheric circulation. It would be like the amount of effect that would be induced on the Mississippi River by dropping in a handful of rocks. There is a LOT of kinetic energy in the global atmospheric circulation, as it extends not just at the surface but through the depth of the atmosphere.

To see where the effect is, take it to the extreme.

What if wind turbines stopped the air. Then we'd have heat without wind. Hot air rises, pulling in more air, etc.

But that;s a hard thing to comprehend.