Adam — The process varies. And each golf course architect has their own ideas. Soil type affects construction. For example, at a pure sand site it is nearly impossible to "shape" a bunker...it just happens, or you might dig it out after the surrounding grass is established.
But typically, on "normal" compactable soils, a small excavator is used (commonly called a mini-excavator). Many will use a smaller bulldozer...D-5 is even too large most times, but I have seen creative shapers use dozers to great success. (I have also seen bunkers built with backhoes and loaders, but this is not what we use.)
The bunker is dug out of a rough shaped area to about 6-8 inches below the desired finish grade. Once it is approved the bunker will be cleaned up to get the edges and depth to the liking of the architect. The area is finished so we can see the final grades...but the bunker floor is still below finished grade. Some will have the bunker edges 100% done at this stage...it depends on the desired edge. If the look is ragged and rough then it probably doesn't get much attention until planting, but if smooth and even, then it may get pre-finished now and then cleaned up later on.
A key step is to set the subgrade (floor) of the bunker. Any excess material needs to be removed and used nearby or hauled away. A decent job in rough shaping will mean less work at this stage...as things are finished.
At this stage we put drainage in — often a sump and drain lines to take water away from the low point, or points. Irrigation follows. At this time the area looks a mess...as if we trashed all the work. But, in reality, trenches have been dug and it all gets put back. Just looks bad.
Then the bunker is finished, usually by hand with whatever edging is called for. The floor is compacted (again) and any lining or stabilization goes in. When the surrounding area is cleaned up (final finishing) and ready for grass it is often the case that sand may be piled in the bunker before planting. Not always. Some work calles for sand to be hauled well after grass is rooted and established.
Once sand is in place it is compacted and any excess removed.
Sod is typically laid around bunkers to stabilize the soils and create a clean edge that seed does not always permit. Of course, if the look is one of ragged edging and a more natural intrusion of grasses on the perimeter, then taller fescuse and seed can be planted (from plugs) or from seed. I think 90% of bunkers get sodded around the edges, most by several feet — 10-ft. or more.
And then there is always the old school method of digging them out later. Mark Fine recently found that several of the classic era architects practiced this method...waiting until the course had been played before deciding their locations.
I think you will get several responses here from designers who do things diferently...it is really a varied process depending on many factors.