I have enabled the Analytics service and connected the logs up to my bot which was a little helpful. I'm able to see 404 errors that look correlated with every event coming from the Slack channel. It's not clear why the communication between the channel (managed by Microsoft) and the bot resource (managed by Microsoft) would have failed internal communications. The logs contain a painfully small amount of detail so it's not clear where to go from here.
I tried connecting it to a completely different Slack bot just to see if there was something wrong with the settings on the bot side and had the same results. I've triple checked that the Slack channel settings are correct.