I have not done any dome work- this sounds like an interesting challenge!
A geodesic dome is actually an assembly of pentagons and hexagons, with triangles filling each "major" shape. Rather than a smooth curve, there is a slight angle at each junction.
Unlike a conventional building, the shape itself is what gives it its' strength, rather than specific structural members. The effect of this is that the only place you can safely place an anchor is dead center where the major shapes come together.
With that in mind, it seems that you'ld either have to run romex, zig-zagging along the frames....or build some sort of trellis to support your stuff. An arched "ladder" made from strut comes to mind. Or, you could be conventional, and "build a box" within the dome, so you can run your pipe in the usual manner.
