Do automated reloaders stack?
Yes. Any augmentation you can buy twice, stacks. If it doesn't stack you will be unable to buy it in a shop, and you'll get scrap instead of it as an event reward.
More precisely, any augmentation with <stackable>true</stackable> in the game resources is stackable and can be bought more than once, and the Automated Re-loader (AUTO_COOLDOWN) is so flagged.
It's known that Scrap Recovery Arms stack additively rather than multiplicatively; other stackable augmentations likely work the same way.