A functionalist is someone who believes in functionalism, in this case referring to the kind defined by Robert A. Heinlein in the short story "The Roads Must Roll". It refers to the belief that a given profession is essential to society in such degree that it should have a larger, or even total share, of the net political power.
In the story, the functionalists were a faction of workers maintaining giant slidewalks that provided nearly all transportation in the US. They shut some of them down to demonstrate their power. The story has a frightening parallel with the views of some modern-day geeks. In the geek node, the subtext seems to be that because geeks know more about an area of technology, they should be in control.
The problem, highlighted within the story, is that just about any profession can claim priority. An argument can be made for hairdressers, waitresses, lawyers, etc. The garbage, sewage, and sanitation workers probably have a better argument than the average geek for being the prime mover that society relies upon.
According to the story, functionalism was a popular belief in the 1920s.