Horizontal drilling is actually very commonplace nowadays. Most all utility lines and pipelines are now installed using directional horizontal drilling rigs to some extent, especially in urban areas.
They drill horizontally under roads, parking lots, buildings, rivers, mountains, etc., and either push or pull thru the bore, pipes, electrical conduits, sleeves, casings, etc.
I often see independent drilling contractors working on the roadside using either Vermeer or Ditch Witch brand drilling rigs, and I suspect it may be possible to strike a good deal with a small contractor to drill a horizontal well. However, I would certainly want to first consult with an expert as to the viability of a horizontal well in its particular location before spending the money on drilling.
Also, I�m assuming that drilling into the side of a mountain or a large hillside is gonna involve drilling thru a lot of rock, and that could be very expensive.
Links to pictures of drilling rigs. They come in many sizes https://i.imgur.com/jHxJcp6.jpg?1 https://i.imgur.com/kSAspGE.png?1 https://i.imgur.com/AQlRhk7.jpg 
