What neuro-motor diseases cause the lower esophageal sphincter (LES) to malfunction?
This is called achalasia. According to UpToDate, the mechanism for this diseases are generally unknown. What is known, however, is that the esophagus has a neural system called the myenteric plexus and if these ganglion cells in the esophagus are damaged in any way (e.g. physically, followed by Wallerian degeneration, or the primary process of degeneration after physical damaging of neurons), it can lead to lower esophageal dysfunction as these neurons regulate the inhibition of the contraction of this sphincter muscle. Dysinhibition leads to a closed sphincter.

Yes, you are right in that the vagus nerve is the primary route of the myenteric plexus to the central nervous system, so damage here might result in dysfunction as well.

Histological studies have reported fewer myenteric neurons in people with achalasia, and inflammatory cells in the region, perhaps suggesting infection or autoimmune etiology 1. 

Additionally, from the UpToDate article, Chagas Disease has been associated with this disease. Chagas disease is a parasitic disease caused by the species Tropanosoma Cruzi, and these parasites multiply within cells, causing them to burst. 

Check out pathophysiology of achalasia on websites such as UpToDate.

1 "Histopathologic features in esophagomyotomy specimens from patients with achalasia." Goldblum JR, Rice TW, Richter JE. Gastroenterology. 1996;111(3):648.

EDIT:

I answered the opposite, but the principles still apply. Here is a wonderful resource on this topic ("Physiology of esophageal motility," http://www.nature.com/gimo/contents/pt1/full/gimo3.html. It includes a diagram showing the vagal neural pathway and the main neurotransmitters responsible for each action (relaxation and contraction).