Doctors have mentioned the vagus nerve to me a couple of times, but they've always been vague, so I'm not sure what they meant.

As far vaccines flaring up achalasia; I'd say possibly. There is a strong theory, with some evidence to support it that achalasia is autoimmune in origin. The theory is that achalasia is an autoimmune response to a virus or bacterial infection. But achalasia develops so slowly and it's not a common illness, by the time it's diagnosed evidence of this is not present. This is not considered conclusive at this point.
(UCLA has some good information about this theory on their website and it's easy to understand. Other sources talk about this as well.) If this is true, vaccines could in theory could flare up achalasia.